UNIVERSITY  OF  CALIFORNIA 
AT   LOS  ANGELES 


TELESCOPE  COMPASS. 


A  MANUAL 


PLANE    SURVEYING; 


CONFINED    TO 


WORK  WITH  THE  COMPASS. 


WITH  AN  APPENDIX. 


AMPLY  ILLUSTRATED. 
REVISED   AND   ENLARGED. 


BY 

THOMAS  BAGOT, 

SUPERINTENDENT   OP   RIPLEY   COUNTY,   INDIANA. 


INDIANAPOLIS,  INDIANA: 

THE  NORMAL  PUBLISHING  HOUSE. 

J.  E.  SHERRILL,  PROPRIETOR. 

1889. 


Entered  according  to  Act  of  Congress  in  the  year  1881, 

BY  J.  E.  SHEERILL, 
In  the  office  of  the  Librarian  of  Congress  at  Washington. 


CARLON  A  HOLLENBECK, 

PRINTERS    AND    BINDERS, 

INDIANAPOLIS. 


TA 


I  DEDICATE  THIS  BOOK  TO  MY  MOTHER 


THE  AUTHOK. 


356 


PREFACE. 


EVERY  person  who  studies  Surveying  from  the  text-books  in  gen- 
eral use,  and  afterward  is  called  upon  to  discharge  the  duties  of  a 
surveyor,  must,  in  the  course  of  time,  become  aware  of  two  things : 
(1)  that  he  has  spent  time  in  learning  much  that  he  has  never 
had,  and  probably  never  will  have,  occasion  to  use,  and  (2)  that  a 
great  deal  he  needs  to  know,  and  must  know,  is  not  to  be  found  in 
the  books. 

This  is  the  case  particularly  under  the  Rectangular  System,  and 
the  author's  experience  has  led  him  to  believe  that  a  necessity  ex- 
ists for  a  book  dealing  directly  with  the  problems  continually 
coming  up  before  surveyors  throughout  the  country,  and  that  such 
a  book  will  be  cordially  received  by  every  person  who  wishes  to 
understand  the  subject  as  it  is  comprehended  in  general  practice. 

And  this,  dear  reader,  accounts  for  the  existence  of  this  little 
book.  You  will  find  it  simply  a  brief  treatise  on  Compass-Sur- 
veying, shorn  of  everything  superfluous,  and  yet  embracing  all 
that  is  necessary  to  a  good  understanding  of  the  subject.  Very 
few  geometrical  or  trigonometrical  terms  are  employed,  and  all 
the  problems  may  be  mastered  by  any  person  having  a  moderately 
good  knowledge  of  arithmetic. 

The  author  does  not  claim  that  the  book  is  above  criticism,  but, 
on  the  contrary,  he  is  well  aware  of  the  fact  that  a  person  disposed 
to  criticise  may  find  in  it  an  ample  field  in  which  to  exercise  his 
talents.  He  trusts,  however,  that  a  search  for  its  faults  will  re- 
sult in  disclosing  enough  merit,  even  among  so  much  demerit,  to 
excuse  him  for  writing  it,  and  so  trusting,  he  submits  it  to  the 
public. 

NEW  MARION,  INDIANA, 
May,  1883. 

(4) 


CONTENTS. 


CHAPTER  I.— INTRODUCTION. 
No.  of  Art. 

(  1).  Surveying  defined. 
(  2).  Branches. 

1.  Topographical  surveying. 

2.  Geodetic  surveying. 

3.  Plane  surveying. 
3).  Measurements,  how 

1.  Actual  area  nearly  always  greater  than  computed  area. 

2.  Smooth  surface  conceived  underneath. 

3.  Impossible  in  many  instances  to  compute  the  area  of 

real  surface. 
(  4).  Corners  defined. 
(  5).  Surveying  instruments  used. 
(  6).  Transit  and  compass. 

1.  Remarks  on  the  transit. 

2.  Remarks  on  the  compass. 
7).  Chain  and  pins. 

(  8).  Chaining  over  hills. 

(  9).  Flag-staff,  drawing  instruments,  etc. 

(10).  Assistants  needed. 

CHAPTER  II.— DESCRIPTION  OP  THE  COMPASS. 

(11).  The  compass  circle,  how  divided. 

(12).  Magnetic  needle  and  center  pin. 

(13).  Degrees,  how  marked.     0°  and  90°  points. 

(14).  Compass  box,  plate,  sights,  etc. 

(5) 


(15).  Description  of  needle.     Delicacy,  how  determine^. 

(16).  Horizontal  angles,  how  measured. 

(17).  Letters  "E"  and  "  W"  reversed  on  compass  face. 

(18).  Eeason  for  this. 

(19).  Kule  for  reading  bearings. 

(20).  Reverse  bearing. 

(21).  Northerly  and  southerly  bearings. 

(22).  Running  lines  east  or  west. 

(23).  Rules  for  measuring  angles. 

1.  When  both  readings  are  in  the  same  quadrant. 

2.  When  one  reading  is  in  each  of  either  the  two  north 

quadrants  or  the  two  south  quadrants. 

3.  When  one  reading  is  in  each  of  either  the  two  east  quad- 

rants or  the  two  west  quadrants. 

4.  When  one  bearing  is  in  each  of  two  opposite  quadrants. 
(24).  Reasons  for  these  rules. 

(25).  Glass  cover  to  compass  box.     Electricity,  how  excited  in  it, 

and  how  removed. 
(26).  Needle  affected  in  other  ways. 
(27).  Sight  compass  and  telescope  compass.     Either  may  be  either 

a  plain  compass  or  a  vernier  compass. 
(28).  Description  of  the  vernier. 
(29).  How  used. 
(30).  Other  kinds  of  verniers. 
(31).  Repairing  the  compass. 

1.  To  re-magnetize  the  needle. 

2.  To  sharpen  the  center  pin. 

3.  To  replace  a  spirit  level. 

4.  To  adjust  a  new  sight. 

5.  To  straighten  the  center  pin. 

6.  To  straighten  the  needle. 

7.  To  put  in  a  new  glass. 

8.  To  regulate  the  movement  of  the  ball. 
(32).  The  compass,  how  carried. 

(33).  Needle  should  be  lifted  from  center  pin  when  the  compass  is 

not  in  use. 
(34)  Compass  should  be  kept  level  when  not  in  use,  and  the  needle 

allowed  to  assume  its  natural  position. 
(35).  The  telescope  compass  gradually  growing  in  favor. 


CONTENTS.  7 

CHAPTER  III.— THE  VARIATION  OF  THE  MAGNETIC  NEEDLE. 

(36).  Meridian  defined. 

(37).  The  true  meridian. 

(38).  Methods  of  determining  the  true  meridian. 

1.  By  a  shadow. 

2.  By  Polaris. 

(a).  Polaris  and  Alioth. 

(b).  Pole  between  them. 

(c ).  The  plumb-line. 

(d).  Greater  accuracy. 

(e ).  When  upper  culmination  occurs  during  the  day. 
(39).  Table  showing  time  of  culmination  of  Polaris. 
(40).  Magnetic  meridian. 
(41).  East  variation  and  west  variation. 
(42).  Agonic  line. 

(43).  Isogonic  lines.  [variation  of  the  needle. 

(44).  The  north  magnetic  pole;  the  effect  of  its  movement  on  the 
(45).  Secular  change  of  variation. 
(46).  Diurnal  change  and  annual  change. 
(47).  Table  showing  diurnal  change  by  hours. 
(48).  Diurnal  and  annual  changes  usually  disregarded  in  practice. 
(49).  Electric  disturbances. 
(50).  The  "dip"  or  inclination  of  the  needle. 
(51).  Many  magnetic  phenomena  imperfectly  understood. 

CHAPTER  IV.— EFFECT  OF  CHANGE  OF  VARIATION  ON  OLD 
LINES,  AND  METHODS  OF  CORRECTING  BEARINGS. 

(52).  Bearings  of  lines  subject  to  constant  change. 

(53).  Illustration. 

(54).  Effect  of  re-surveying  lines  without  considering  the  change. 

(55).  Area  of  tract  not  affected. 

(56).  To  determine  the  present  bearing  of  a  line. 

1.  Bearing  at  time  of  a  previous  survey. 

2.  Date  of  previous  survey. 

3.  Annual  amount  of  secular  change.     This  may  be  de- 

termined in  various  ways. 
(1).  By  comparison  of  bearings. 
(2).  By  establishing  a  true  meridian. 
(3).  By  interpolation. 


O  CONTENTS. 

(57).  Table  showing  the  variation  of  the  needle  at  important  sta- 
tions. 

(58).  Table  showing  annual  amount  of  secular  change  in  certain 
localities. 

(59).  Tables  to  be  used  in  approximation. 

(60).  Determining  variation  for  past  times. 

(61).  Advantage  of  basing  bearings  on  the  true  meridian. 

(62).  Magnetic  meridian  bearings  subject  to  constant  change. 

(63).  To  determine  the  true  bearing  of  a  line  from  its  magnetic 
bearing. 

1.  When  the  variation  is  west. 

2.  When  the  variation  is  east. 
(64).  Bearings  to  be  changed. 

(65).  When  supplement  is  to  be  taken. 

(65).  Changing  from  the  true  meridian  to  the  magnetic  meridian. 

CHAPTER  V.— METHOD  OF  RUNNING  LINES. 

(67).  Finding  a  corner,  etc. 

(68).  Determining  position  of  corner  from  witnesses. 

(69).  Where  trees  are  not  available  for  witnesses,  other  things  are 
used. 

(70).  Method  of  describing  witness  trees. 

(71).  Starting  the  survey  of  a  line. 

(72).  Course  and  distance  of  line  must  at  least  be  known  approxi- 
mately before  the  line  can  be  surveyed. 

(73).  Setting  the  compass  and  measuring  the  line. 

(74).  Setting  off  variation  on  the  vernier. 

1.  When  the  variation  is  east. 

2.  WThen  the  variation  is  west. 
(75).  Surveying  the  line. 

(76).  How  chained. 
(77).  Pins,  stakes,  etc. 

(78).  Relative  places  on  line  of  men  engaged  in  the  survey. 
(79).  Continuation  of  the  line  to  the  opposite  corner. 
(80).  In  case  the  line  does  not  strike  the  corner. 
(81).  Illustration. 

(82).  Rules  for  correcting  the  stakes. 

(83).  Terminations  of  lines  and  starting  point  regarded  as  vertices 
of  an  isosceles  triangle. 


(84).  Examples  in  moving  (correcting)  the  stakes. 

(85).  Abbreviations  used. 

(86).  Errors  caused  by  imperfection  of  instruments,  etc. 

(87).  Correction  of  assumed  bearing. 

(88).  Rules  for  correcting  bearings. 

1st  method.     Derivation  of  rule.    - 
(1).  Trigonometrical  lines. 
(2).  Sine  and  cosine  defined. 

(3).  Study  of  relation  of  sine  and  cosine  important. 
(4).  Application  of  these  lines. 
(5).  -When  the  angle  is  large. 
2d  method. 

(1).  Modification  of  first  method. 
(2).  Illustration. 
3d  method. 
4th  method. 

(89).  Examples  of  lines  whose  bearings  are  to  be  corrected. 
(90).  Is  amount  of  correction  to  be  added  or  subtracted? 

1.  Rule  for  north-east  and  south-west  courses. 

2.  Rule  for  north-west  and  south-east  courses. 
(91).  Examples  under  these  rules. 

(92).  Thus  far,  all  bearings  have  been  based  on  the  true  meridian. 
(93).  Bearings  of  lines  based  on  the  magnetic  meridian. 

1.  Rule  for  correction. 

2.  Demonstration  of  the  rule. 

(94).  Rule  good  while  the  needle  moves  westward. 

(95).  Bearings  to  be  corrected. 

(96).  Abbreviations,  etc. 

(97).  Completion  of  the  survey. 

(98).  Assistants  generally  sworn. 

(99).  Instruments  should  be  tested  frequently. 

(100).  Backsights. 

CHAPTER  VI.— UNITED  STATES  RECTANGULAR  SURVEYING. 

(101).  Public  lands,  how  divided. 

(102).  Townships,  how  divided. 

(103).  These  provisions  sufficient. 

(104).  Fundamental  lines  and  initial  point. 

(105).  Some  imperishable  mark  chosen  for  the  initial  point. 


10  CONTENTS. 

(106).  Survey  of  range  lines  or  meridians,  etc. 

(107).  Survey  of  parallels,  etc. 

(108)".  Convergence  of  meridians.     Correction  lines. 

(109).  Auxiliary  meridians,  etc. 

(110).  Survey  of  townships  north  of  the  base-line  and  east  of  the 

principal  meridian. 
(111).  Survey  of  townships  north  of  the  base-line  and  west  of  the 

principal  meridian,  etc. 
(112).  Excesses  and  deficiencies. 
(113).  Congressional  township  and  civil  township. 
(114).  Survey  of  sections. 

1.  Preliminaries. 

2.  How  to  obtain  bearings. 

3.  Sections,  how  numbered. 

4.  Survey  of  section  36. 

5.  Survey  of  the  rest  of  eastern  tier  of  sections. 

6.  Completion  of  the  township. 

(115).  Full  sections  and  fractional  sections.  The  "double  frac- 
tional." 

(116).  Meander  corners  and  meander  lines. 

(117).  Making  up  the  field-notes. 

(118).  Monuments  adapted  to  the  country  surveyed. 

(119).  Work  of  Government  deputy  extends  only  to  the  division  of 
the  township  into  sections. 

(120).  Advantages  of  the  Rectangular  system,  etc. 

CHAPTER  VII.— THE  DIVISION  AND  SUBDIVISION  OF  THE 
SECTION. 

(121).  Subsequent  surveys  must  be  made  in  accordance  with  the 

original. 

(122).  The  ideal  section  and  the  real  section. 
(123).  The  divisions  and  principal  sub-divisions  of  the  section. 
(124).  Names  of  these  divisions  and  sub-divisions. 
(125).  Descriptions  of  land  generally  qualified  by  the  words  "more 

or  less." 
(126).  Corners,  how  named. 

1.  Section  corners. 

2.  Quarter-section  corners. 

3.  Half-quarter  corners. 

4.  Fourth-quarter  corners. 


CONTENTS.  11 

(127).  Section  lines  and  center  lines. 
(128).  Position  of  corners,  how  determined. 

1.  Section  corners  and  exterior  quarter-section  corners. 

2.  Center  corners,  methods  of  setting. 

(1).  By  crossing  the  center  lines. 

(2).  By  bisecting  east  and  west  center  line. 

3.  Half-quarter  corners. 

4.  Fourth-quarter  corners. 

5.  Other  corners. 

(129).  Examples  in  setting  corners. 

(130).  Survey  of  the  divisions  and  sub-divisions  of  the  section: 

1.  Quarter  section. 

2.  Half-quarter  section. 

3.  Fourth-quarter  section. 
(131).  General  rule. 

(132).  When  some  of  the  boundary  lines  are  known. 

(133).  Tracts  to  be  surveyed. 

(134).  Independent  corners,  lines,  and  tracts. 

(135).  Illustration  of  independent  corners,  etc. 

(136).  Dependent  corners,  lines,  and  tracts. 

(137).  Dependent  lines,  etc.,  how  surveyed. 

(138).  Examples  of  dependent  tracts. 

(139).  Description  by  "metes  and  bounds." 

(140).  Rules  for  setting  corners,  etc.,  in  full  sections,  generally 

apply  to  fractional  sections. 
(141).  When  a  tract  of  land  lies  partly  in  one  section  and  partly 

in  another. 

CHAPTER  VIII.— FIELD-NOTES. 

(142).  Field-notes  of  sectional  survey  by  the  Government  deputy. 
(143).  Contents  of  full  section,  supposition  regarding  it,  etc. 
(144).  Plot  of  township. 
(145).  Explanation  of  the  plot. 
(146).  List  of  witnesses  to  corners. 

1.  Exterior  corners. 

2.  Interior  corners. 
(147).  Other  particulars. 

1.  Area  of  fractional  quarters. 

2.  Creeks,  etc. 

3.  Offsets. 


12  CONTENTS. 

(148).  Surveyor  needs  copy  of  original  field-notes. 
(149).  What  surveyors'  records  should  contain. 
(150).  Pocket  record. 
(151).  Description  of  pocket  record. 

1.  Left-hand  page. 

2.  Right-hand  page. 

(152).  Instructions  regarding  bearings. 
(153).  Representation  of  surveys  on  the  plot. 
(154).  Approximating  the  bearing  of  a  line. 
(155).  Illustration. 

(156).  Principles  underlying  the  method. 
(157).  The  field-book. 

(158).  Method  of  keeping  the  field-book  in  independent  surveys. 
(159).  When  new  witnesses  should  be  taken. 
(160).  Method  of  keeping  the  field-book  in  dependent  surveys. 
(161).  Names  of  stations,  witnesses,  etc. 

(162).  Records  by  authorized  surveyors  taken  as  prima  facie  evi- 
dence in  favor  of  surveys  recorded. 
(163).  Other  methods  of  keeping  field-notes. 

CHAPTER  IX.— RE-LOCATION  OF  CORNERS. 

(164).  Trouble  caused  by  lost  corners. 

(165).  Nature  of  this  trouble.     Means  of  re-locating  corners. 

1.  Remains  of  missing  corner  or  witnesses. 

2.  By  course  and  distance  from  some  other  corner. 

3.  By  retracing  old  line  by  marks  on  trees,  etc. 

4.  By  projecting  lines. 

(1).  Illustration  of  this  method. 

(2).  The  reverse  of  the  method  by  which  the  corners 

were  established. 
(3).  Another  illustration. 
(4).  Examples  for  practice. 

5.  A  last  resort. 

(1).  Setting  S.  }  corner. 

(2). 'May  not  agree  in  position  with  corner  lost. 

(S).  A  case  illustrated. 

(4).  Lost  section  corner. 

(5).  Case  of  disagreement  illustrated. 

(6).  When  a  quarter  corner  can  not  be  found. 


CONTENTS.  13 

(166).  Re-locating   original   corners   to  the  variable  quarters  of 

fractional  sections. 

(167).  To  set  a  quarter  corner  between  two  fractional  sections. 
(168).  To  set  an  exterior  corner  to  a  fractional  section,  or  to  any 

exterior  section. 

1.  When  there  is  an  offset. 

2.  When  there  is  no  offset. 
(169).  Examples. 

(170).  He-location  of  subsequent  corners,  etc. 

CHAPTER  X.— DESCRIPTIONS  OF  LAND. 

(171).  Necessity  of  a  good  description. 

(172).  Length  of  lines  and  area  of  tracts  should  be  given  in  sur- 
veyor's measure. 

(173).  Tables  of  equivalents. 

(174).  Examples  for  reduction. 

(175).  Fractions  of  a  chain  expressed  in  links. 

(176).  General  rule  for  reduction. 

(177).  Area  of  tracts,  how  expressed. 

(178).  Description  of  independent  tract. 

(179).  Examples  of  errors  in  descriptions. 

(180).  Examples  for  correction. 

(181).  Erroneous  descriptions. 

(182).  Use  the  words  "  more  or  less." 

(183).  In  describing  dependent  tracts  the  course  and  distance  of 
each  boundary  should  generally  be  given. 

(184).  The  description  should  state  whether  the  bearings  are  based 
on  the  true  meridian  or  on  the  magnetic  meridian. 

(185).  Lines  running  north,  etc. 

(186).  Surveys  generally  made  in  accordance  with  the  description. 

CHAPTER  XL— OBSTACLES  TO  ALIGNMENT  AND  MEASUREMENT. 

(187).  Obstacles  met  on  line. 

(188).  Two  classes  of  obstacles. 

(189).  Methods  of  spanning  obstacles  of  first  class. 

1.  By  perpendiculars. 

2.  By  an  equilateral  triangle. 

3.  By  a  right-angled  triangle.  [urement. 

(1).  When  an  obstacle  both  to  alignment  and  meas- 
(2).  When  an  obstacle  to  measurement  alone. 


14  COXTENTS. 

4.  By  symmetrical  triangles. 

5.  When  a  fence  is  built  on  or  near  the  line. 

(1).  When  offset  line  terminates  on  the  opposite  side. 

(2).  When  the  offset  line  terminates  on  the  same  side. 

(a).  When  the  distance  missed  is  greater  than 

the  offset. 

(b).  When  the  offset  is  greater  than  the  distance 
missed. 

6.  Surveying  over  hills. 

(190).  Methods  of  spanning  obstacles  of  the  second  class. 

1.  By  a  right-angled  triangle. 

2.  By  symmetrical  triangles. 

3.  By  similar  triangles. 
(191).  Other  methods,  etc. 


CHAPTER  XII.— COMPUTATION  OP  AREA. 

(192).  Advantage  of  expressing  dimensions  of  tracts  in  chains  and 

links. 

(193).  Special  rules  deduced. 
(194).  Examples  in  computation. 
(195).  Every  tract  of  land  a  polygon  in  shape. 
(196).  Rectangles. 
(197).  Parallelograms. 
(198).  Trapezoids. 
(199).  Triangles. 

1.  With  base  and  altitude  given. 

2.  With  no  altitude  given. 
(200).  Trapeziums. 

(201).  Any  figure. 

(202).  Computation  of  area  by  latitudes  and  departures. 

(203).  Latitude  and  departure,  as  applied  to  courses,  defined. 

(204).  North  and  south  latitudes,  and  east  and  west  longitudes. 

(205).  Signs  of  latitudes  and  departures. 

(206).  Relation  of  sine  and  cosine  to  departure  and  latitude. 

(207).  Table  of  natural  sines  and  cosines  explained. 

(208).  Examples  in  computing  latitudes  and  departures. 

(209).  Columns  marked  differently  at  top  and  bottom. 

(210).  Traverse  tables. 


CONTENTS.  16 

(211).  Sum  of  latitudes  and  sum  of  departures  in  every  cor- 
rect survey. 

(212).  A  trial  survey. 

(213).  Explanation. 

(214).  When  a  discrepancy  exists. 

(215).  Initial  line  or  meridian. 

(216).  Difference  between  longitudes  and  departures. 

(217).  How  to  determine  the  longitude  of  a  course. 

(218).  Algebraic  sum  must  be  used. 

(219).  Simplification  of  the  rule. 

(220).  Computation  of  area  by  longitudes. 

(221).  North  products  and  south  products. 

(222).  Double  longitudes. 

(223).  Method  of  keeping  the  data. 

(224).  General  rule  for  computing  areas  by  double  longitudes. 

(225).  To  determine  the  most  westerly  corner. 

(226).  Examples  in  computation. 

(227).  Courses  without  departures,  etc. 

(228).  Not  absolutely  necessary  that  the  meridian  be  drawn 
through  the  most  westerly  station. 

CHAPTER  XIII.— LAYING  OUT  AND  DIVIDING  UP  LAND. 

(229).  No  general  rule  can  be  given. 

(230).  What  are  known  in  problems  to  be  considered. 

(231).  To  lay  out  a  square. 

(232).  To  lay  out  a  rectangle. 

(233).  To  lay  out  a  parallelogram. 

(234).  To  lay  out  a  right-angled  triangle. 

(235).  To  lay  out  a  trapezoid. 

(236).  To  lay  off  any  figure. 

(237).  Things  to  be  considered  in  making  partition. 

(238).  Nature  of  problems  chosen. 

(239).  To  divide  a  rectangle  into  equal  parts. 

(240).  To  divide  a  rectangle  into  unequal  parts. 

(241).  Problems. 

CHAPTER  XIV.— SURVEYING  TOWN  LOTS. 

(242).  Description  of  town  lots,  blocks,  etc. 
(243).  Survey  of  town,  how  based. 


16  CONTENTS. 

(244).  Usual  shape  of  lots,  etc. 

(245).  What  the  plot  of  a  town  should  show. 

(246).  Particulars  respecting  Figure  58. 

(247).  Illustration  of  a  town  in  which  the  lots  vary  in  size. 

(248).  Method  of  surveying  lot  number  11  in  the  figure. 

(249).  Examples  for  practice. 

(250).  Chain  or  tape  used  in  surveying  should  be  tested  frequently. 

CHAPTER  XV.— PLOTTING. 

(251).  Plotting  defined. 
(252).  Instruments  used. 

1.  Drawing  board. 

2.  T-square. 

3.  Euler. 

4.  Drawing  pen. 

5.  Dividers  or  compasses. 

6.  Protractor. 

7.  Diagonal  scale  of  equal  parts. 

(253).  Units  of  the  scale  may  have  various  equivalents. 

(254).  Plotting  bearings. 

(2551.  Examples  for  practice. 

(256).  Plotting  rectangular  tracts. 

(257).  Plotting  tracts  in  general. 

(258).  A  particular  case.    When  a  survey  does  not  "close." 

(259).  The  pantograph. 

(260).  Locating  objects  on  the  plot. 

(261).  Coloring  plots  and  maps. 

CHAPTER  XVI.— SURVEYING  WITHOUT  A  COMPASS. 

(262).  The  compass  nearly  always  necessary. 

(263).  Setting  corners. 

(264).  Establishing  lines. 

(265).  Setting  out  perpendiculars. 

(266).  Survey  of  rectangular  tracts. 

(267).  Measurements. 

APPENDIX. 
Land  Decisions. 
Table  of  natural  sines  and  cosines. 


MANUAL 


PLANE  SURVEYING. 


CHAPTER  I. 

INTRODUCTION. 

ART.  ( 1 ).  SURVEYING  is  that  branch  of  applied  mathematics 
which  embraces  operations  for  finding,  (1)  the  relative  positions 
of  points  on  the  earth's  surace,  (2)  the  area  of  any  portion  of  its 
surface,  and  (3)  the  contour  or  shape  of  any  part  of  its  surface,  so 
that  it  may  be  represented  in  maps  and  plots. 

(  2  ).   It  is  divided  into  three  branches  : 

1.  Topographical  Surveying,  or  Topography,  includes  operations  for 
determining  the  contour  of  portions  of  the  earth's  surface  and  re- 
presenting it  on  paper. 

2.  Geodetic  Surveying,  or  Geodesy,  takes  into  consideration  the 
curvature  of  the  earth's  surface  and  is  employed  in  extensive  sur- 
veys. 

3.  Plane  Surveying  does  not  regard  the  curvature"  of  the  earth's 
surface  and  all  lines  are  measured  as  on  a  plane.     It  is  used  in  lo- 
cal work. 

(3).  All  measurements  in  surveying  are  made  as  nearly  hori- 
zontally as  possible,  and  the  area  of  a  tract  of  land  is  not  its  ac- 
tual surface  measure,  unless  the  tract  be  perfectly  level,  but  the 
amount  of  land  enclosed  by  its  boundaries  measured  horizontally, 
instead  of  with  the  inclinations  of  the  surface  over  which  they  run. 
2  (17) 


18  MANUAL   OF  PLANE  SURVEYING. 

1.  The  actual  area,  therefore,  is  nearly  always  greater  than  the 
computed  area,  and  increases  in  proportion  to  the  inequality  of 
the  surface. 

2.  We  may  conceive  a  smooth  surface  at  the  level  of  the  ocean 
underlying  the  surface  of  the  land;  then  the  area  of  a  tract  of 
land  is  equal  to  the  contents  of  a  figure  formed  by  projecting  the 
boundaries  of  the  tract  on  the  horizontal  surface  below. 

3.  Were  the  real  surface  considered,  it  would  be  impossible  in 
many  instances  either  to  compute  its  area  or  represent  its  figure 
on  paper. 

( 4 ).  The  extremities  of  lines  in  surveying  are  called  corners, 
and  each  corner  marks  the  vertex  of  an  angle  formed  by  the  meet- 
ing of  two  lines.  The  corner  is  a  mathematical  point,  and  may 
or  may  not  be  marked  with  a  monument. 

(  5  ).  Lines  are  surveyed  either  with  a  solar  or  a  magnetic  in- 
strument. With  the  former,  where  more  precision  than  expedi- 
tion is  required ;  and  with  the  latter,  where  expedition  is  of  more 
importance  than  precision. 

(  6 ).  The  principal  magnetic  instruments  in  use  are  the  transit 
and  compass. 

1.  The  transit  is  provided  with  a  telescope,  and  at  the  present 
time  is  so  constructed  as  to  be  adapted  to  the  measurement  of 
both  horizontal  and  vertical  angles.     It  serves  many  important 
purposes  independent  of  the  assistance  of  the  magnetic  needle,  and 
is  not  strictly  a  magnetic  instrument. 

2,  The  compass  is  either  supplied  with  sights  or  a  telescope, 
and  is  strictly  a  magnetic  instrument.     It  is  not  usually  adapted 
to  measuring  vertical  angles.     The  lightness,  simplicity,  and  con- 
venience of  the  compass  have  brought  it  into  almost  general  use 
in  common  surveying,  and  the  following  chapter  is  devoted  to  a 
description  of  it.     The  transit  may  be  found  described  in  almost 
any  comprehensive  work  on  Surveying. 

(  7 ).  Measurements  of  lines  in  surveying  are  made  with  an 
iron  or  steel  chain*  usually  33  feet  or  2  rods  long  and  divided 
into  50  links.  It  has  a  handle  at  each  end  by  which  it  is  carried 
during  the  survey,  and  the  successive  chain-lengths  are  maked 

*For  convenience  in  counting  the  links,  this  chain  is  divided  into  five 
parts  of  ten  links  each  by  four  brass  or  copper  tags.  The  real  chain  is  100 
links  or  66  feet  in  length,  and  called  a  Gunter's  chain,  from  its  inventor. 
Ihe  half-chain  of  50  links  is  used  for  convenience.  A  link  is  7.92  inches  in 
length.  In  Government  surveys,  a  chain  66.06  feet  in  length  is  used.  The 
M  foot  being  added  to  make  up  for  "  slack,"  etc. 


MANUAL   OF  PLANE   SURVEYING.  19 

with  pins  of  iron  or  steel  wire,  generally  10  or  12  inches  in  length, 
sharpened  at  one  end  and  bent  into  the  form  of  a  ring  at  the'other. 

In  the  ring  is  sometimes  tied  a  bright  ribbon  or  piece  of  cloth  to 
render  the  pin  more  conspicuous,  in  order  that  it  may  be  easily 
found  by  the  person  who  carries  the  rear  end  of  the  chain,  and 
such  colors  should  always  be  chosen  as  will  contrast  most  with 
the  surface  to  be  surveyed. 

(  8  ).  In  chaining  up  and  down  hill,  the  chain  must  be  kept  taut 
and  horizontal,  as  on  a  level  surface.  In  order  to  do  this,  it  is 
sometimes  necessary  to  drop  the  pin  from  the  front  end  of  the  chain, 
or  elevate  the  rear  end  of  the  chain  to  a  point  exactly  over  the 
pin  that  sticks  in  the  ground. 

(9).  A  straight  staff,  about  1£  inches  in  diameter  and  8  or  10 
feet  high,  surmounted  by  a  small  flag  of  brilliant  color,  is  used  in 
alignment;  and  a  good  set  of  drawing  instruments,  drawing  board, 
t-square,  triangle,  protractor,  ruler,  etc.,  are  necessary  in  drawing 
and  plotting. 

(10).  In  field-work  the  surveyor  generally  needs  four  assist- 
ants ;  two  chain-men,  one  flag-man,  and  a  marker.  The  first  two 
measure  the  line  with  the  chain,  the  third  carries  the  flag,  and 
the  fourth  assists  the  chain-men  by  marking  the  line  with  stakes 
at  the  proper  distances.  To  this  force  it  is  sometimes  necessary 
to  add  one  or  more  ax-men,  as  bushes  may  have  to  be  cut  out  of 
the  way  and  trees  marked. 

QUESTIONS  ON  CHAPTER  I. 

1.  Define  Surveying. 

2.  Into  what  three  branches  is  it  divided? 

3.  What  does  each  branch  embrace  ? 

4.  How  are  measurements  made  in  surveying? 

5.  Why  is  the  actual  area  of  a  tract  of  land  nearly  always 

greater  than  its  computed  area? 

6.  What  is  a  corner? 

7.  Name  the  two  kinds  of  instruments  used  in  surveying. 

8.  Name  the  principal  magnetic  instruments. 

9.  Describe  the  transit.     The  compass. 

10.  How  are  lines  measured  in  surveying? 

11.  How  should  the  chain  be  held  in  chaining  over  hills? 

12.  How  many  assistants  does  the  surveyor  usually  need? 


CHAPTER  II. 

DESCRIPTION  OF  THE  COMPAS& 

(11).  The  compass  consists  essentially  of  the  compass  circle, 
the  magnetic  needle,  and  the  sights.  The  circle  has  its  circum- 
ference raised  and  divided  into  360  equal  parts  or  degrees,  and 
these  are  usually  subdivided  to  half  or  quarter  degrees. 

( 12  ).  At  the  center  of  the  compass  circle"  is  placed  a  perpen- 
dicular pin,  called  the  center  pin.  Upon  this  pin  the  magnetic 
needle  is  balanced  in  such  a  way  as  to  mark  opposite  points  of  tht 
divided  circumference  of  the  circle. 

( 13 ).  The  degrees  on  the  circumference  are  marked  from  twa 
opposite  points,  called  0  points,  up  to  90°  to  the  right  and  left. 
The  90°  points  are,  therefore,  opposite  one  another  also.  One  of 
the  0  points  is  called  the  north  point,  and  the  other  the  south 
point,  and  one  of  the  90°  points  is  called  the  east  point,  and  the 
other  the  west  point. 

( 14).  The  compass  circle  is  enclosed  in  what  is  known  as  the 
compass  box,  and  this  rests  upon  the  compass  plate.  The  box  is 
usually  between  5  and  7  inches  in  diameter,  and  the  plate  about 
14  or  16  inches  long.  At  the  ends  of  the  plate  perpendicular 
sights  are  placed.  These  sights  have  slits  in  them,  and  are  so 
placed  that  the  line  of  sight  from  one  of  them  to  the  other  will 
strike  opposite  points  of  the  graduated  circumference  of  the  com- 
pass circle.  Between  the  compass  box  and  the  sights  are  usually 
placed  two  spirit  levels,  one  at  right  angles  to  the  other.  These 
are  used  in  leveling  the  compass.  The  compass  rests  upon  a  tri- 
pod or  Jacob's  staff',  at  the  head  of  which  is  a  ball  and  socket 
joint,  enabling  a  person  to  move  the  compass  as  he  may  wish. 

(15).  The  needle  is  a  magnetized  steel  bar,  very  delicately 

(20) 


MANUAL   OF   PLANE   SURVEYING.  21 

balanced  upon  the  point  of  the  center  pin,  and  it  is  a  little  shorter 
than  the  diameter  of  the  compass  circle.  The  delicacy  of  the 
needle  is  determined  by  the  number  of  horizontal  vibrations  it 
will  make  before  coming  to  rest  after  being  disturbed.  Needles 
5  or  5 2  inches  in  length  are  generally  preferred  by  surveyors  to 
longer  or  shorter  ones. 

(  16 ).  Horizontal  angles  are  measured  with  the  compass  by 
turning  the  sights  from  one  line  of  the  angle  to  the  other  and 
noting  the  number  of  degrees  passed  over  by  the  end  of  the  needle. 
The  sights  are  sometimes  arranged  for  the  measurement  of  verti- 
cal angles  also. 

( 17  ).  The  letters  "E"  and  "  W"  are  reversed  on  the  compass 
face,  but  it  will  be  plainly  seen  that  the  arrangement  enables  the 
surveyer  to  take  the  direction  (bearing)  of  a  line  more  readily 
from  the  compass  face,  and  reduces  the  liability  to  err  in  the 
reading. 

( 18 ).   This  may  be  illustrated  by  Fig.  1. 


FIG.  1. 

Suppose  the  sights  set  in  the  direction  of  the  0  points  of  the 
circle,  and  that  the  compass  be  turned  until  the  north  end  of  the 
needle  marks  the  point  midway  between  the  north  point  (gener- 


22  MANUAL   OF   PLANE   SURVEYING. 

ally  marked  with  afleur  de  Its,  instead  of  the  letter  N)  and  E,  or 
to  45°.  The  sights  have  moved  in  the  direction  of  the  second 
hand  of  a  watch,  and  the  bearing  of  the  line  marked  by  them  is 
N  45°  E,  which  is  read  directly  from  the  north  end  of  the  needle. 

( 19  ).   The  following  is  the  general  rule  for  all  readings: 

Note  the  letters  between  which  the  end  of  the  needle  comes,  and 
to  what  number.  Then  name  the  letter  N  or  S  (as  the  case  may 
be)  that  is  the  nearer  to  the  end  of  the  needle  from  which  you  are 
reading,  next  the  number  of  degrees  to  which  the  needle  points, 
and  lastly  the  letter  E  or  W  that  is  the  nearer  to  the  same  end  of 
the  needle. 

(  20 ).  If  the  preceding  reading  had  been  taken  from  the  south 
end  of  the  needle,  the  bearing  would  have  been  S  45°  W,  the  re- 
verse of  X  45°  E,  but  equivalent  to  it,  and  indicating  the  bearing 
taken  from  the  opposite  end  of  the  line.  S  45°  W  is  called  a 
southerly  bearing,  and  N  45°  E  is  called  a  northerly  bearing. 

(21).  No  matter  how  the  lines  run,  the  same  end  of  the  com- 
pass (the  north  end  is  preferred)  should  be  kept  in  front. 

(22).  In  running  lines  east,  the  E  point  of  the  compass  circle 
will  be  turned  toward  the  north,  and  in  running  west,  the  W 
point  will  be  turned  north.  All  bearings  should  be  read  from  the 
north  end  of  the  needle. 

(23 ).   In  measuring  angles  observe  the  following  rules: 

1.  When  both  readings  are  in  the  same  quadrant,  as  between 
N  and  E,  N  and  W,  S  and  E,  or  S  and  W,  the  angle  is  equal  to 
the  difference  between  the  two  readings.     Thus,  the  angle  between 
N  56°  E  and  N.  43°  E  is  equal  to  13°. 

2.  When  one  reading  is  in  each  of  either  the  two  north  quad- 
rants or  the  two  south  quadrants,  the  included  angle  is  equal  to 
the  sum  of  the  two  readings.     Thus,  the  included  angle  of  N  35° 
E  and  N  23°  W  is  equal  to  35°  +  23°  =  58°. 

3.  When  one  reading  is  in  each  of  either  the  two  east  quadrants 
or  the  two  west  quadrants,  the  included  angle  is  equal  to  the  sum 
of  the  readings  subtracted  from  180°.     Thus,  the  angle  included 
between  N  50°  E  and  S  37°  E  is  equal  to  180°— (50°  +  37°)  =  93°. 

4.  When  one  reading  is  in  each  of  two  opposite  quadrants,  the 
angle  is  equal  to  the  difference  of  the  readings  subtracted  from 


MANUAL    OF    PLANE   SURVEYING. 


180°.     Thus,  the  included  angle  of  N  16°  E  and  S  12°  W  is  equal 
to  180°  —  (16°  —  12°)  =  176°. 

(  24:).   The  reasons  for  these  rules  will  be  seen  in  Fig.  2. 


FIG.  2. 

The  angle  included  between  the  courses  A  B  and  A  C  is  equal 
to  59°  —  28°  =  31°,  as  both  readings  are  in  the  same  quadrant. 

The  angle  included  between  the  courses  A  B  and  A  D  is  equal 
to  28°  +  36°  =  64°,  because  one  reading  is  in  each  of  the  two 
north  quadrants. 

The  angle  included  between  the  courses  A  C  and  A  F  is  equal 
to  180°  —  (59°  +  33°)  =  88°,  because  one  is  in  each  of  the  two 
east  quadrants. 

The  angle  included  between  the  courses  A  D  and  A  F  is  equal 
to  180°  —  (36°  —  33°)  =  177°,  because  the  courses  are  in  opposite 
quadrants. 

(  25 ).  The  compass  box  is  protected  by  a  glass  covering  over 
which  fits  a  brass  lid.  Care  should  be  taken  while  the  brass  lid 
is  off  that  no  electricity  be  excited  in  the  glass  by  the  friction  of 
the  hand  or  a  cloth  upon  its  surface,  as  it  interferes  with  the  work- 


24  MANUAL   OF   PLANE   SURVEYING. 

ing  of  the  needle  and  may  cause  a  serious  error.  However,  when 
the  fluid  does  exist,  it  may  be  removed  by  breathing  on  the  glass 
or  touching  it  in  various  places  with  the  moistened  finger. 

(  26  ).  The  action  of  the  needle  is  also  affected  by  pieces  of  iron 
or  steel  brought  or  kept  near  it.  This  materially  interferes  with 
its  use  at  sea,  particularly  on  iron  ships.  While  surveying,  noth- 
ing having  a  tendency  to  affect  the  action  of  the  needle  should  be 
carried  upon  the  person  or  allowed  near  the  compass. 

(  27  ).  Two  kinds  of  compasses  are  in  use — the  sight  compass 
and  the  telescope  compass.  Each  of  these  may  be  either  a  plain 
compass  or  a  vernier  compass.  The  plain  compass  is  not  very  ex- 
tensively used,  as  all  readings  are  made  from  its  face  alone,  and 
can  not  be  depended  on  for  precision.  In  the  plain  compass,  the 
line  of  sight  lies  in  the  direction  of  the  0  points  of  the  compass 
circle.  The  vernier  compass  differs  from  the  plain  compass  in 
having  its  compass  circle,  to  which  a  "  vernier"  is  attached,  mov- 
able, generally  through  a  short  arc,  about  its  center,  thus  enabling 
the  surveyor  to  set  the  zeros  or  0  points  of  the  circle  at  an  angle 
with  the  line  of  sight.  This  angle  is  read  from  the  vernier.  The 
movement  of  the  circle  is  effected  by  means  of  a  thumb-screw  that 
gives  it  a  slow  motion.  When  the  required  angle  is  set  off,  the 
vernier  is  clamped  to  the  plate  of  the  compass  and  the  readings 
taken.  The  vernier  enables  a  surveyor  to  take  a  certain  class 
of  readings  "closer "or  with  greater  precision  than  is  possible 
without  it,  as  it  gives  him  the  advantage  of  a  double  index ; 
yet  readings  down  to  a  very  small  angle,  say  V  or  30",  are 
hardly  ever  trustworthy,  owing  to  the  difficulty  a  surveyor  meets  in 
setting  the  needle  to  exactity.  The  fault,  however,  is  not  in 
the  vernier. 

(  28  ).  The  VERNIER*  consists  of  an  arc  divided  into  a  certain 
number  of  equal  parts,  and  moving  within  another  arc  whose  di- 
visions are  somewhat  larger  or  smaller  than  its  own.  The  first 
arc  (vernier),  as  stated  before,  is  attached  to  the  compass  circle 
and  moves  with  it  around  a  common  center ;  the  second  arc  is 
called  the  "  limb,"  and  is  generally  on  the  brass  plate  of  the  com- 
pass upon  which  the  circle  moves,  so  that  the  outer  edge  of  the 
vernier  coincides  with  the  inner  edge  of  the  limb. 

*The  vernier  is  so  called  from  its  inventor,  and  on  this  account  the  word 
is  sometimes  written  with  an  initial  capital.  It  was  first  applied  to  the 
compass  by  David  Rittenhouse  of  Philadelphia. 


MANUAL   OF  PLANE   SURVEYING.  2D 

(  29).  Let  us  now  suppose  the  divisions  on  the  limb  to  equal 
half-degrees,  and  that  the  vernier-arc,  corresponding  to  twenty- 
nine  divisions  of  the  limb,  is  divided  into  thirty  equal  parts. 

It  is  plain,  since  30  divisions  of  the  vernier  equal  29  divisions 
of  the  limb,  that  one  division  of  the  limb  equals  ?§  divisions  of 
the  vernier,  and  that  one  division  of  the  vernier  equals  f§  di- 
vision of  the  limb. 

But  each  division  of  the  limb  equals  30'  or  one-half  of  one  de- 

OA/\xOQ 

gree  ;  therefore,  one  division  of  the  vernier  will  equal  — HQ —  =  ^9'  > 
which  is  one  minute  less  than  a  division  of  the  limb. 

Now,  suppose  the  zero  of  the  vernier  to  correspond  with  the 
zero  of  the  limb ;  then  the  0  points  of  the  compass  circle  lie  in  the 
line  of  sight.  If  now  we  turn  the  vernier  until  its  first  division 
from  zero  coincides  with  the  first  division  from  zero  on  the  limb, 
and  on  the  same  side  of  zero  as  the  division  of  the  vernier,  the  0 
points  of  the  compass  will  make  an  angle  of  V  with  the  line  of 
sight;  if  the  second  division  of  each  coincide,  the  angle  will  be  2'; 
if  the  third,  it  will  be  3',  and  so  on  by  the  same  increase,  so  that 
if  we  make  the  twenty-ninth  division  of  each  correspond,  the  an- 
gle will  be  29' ;  and  if  we  turn  still  further  until  the  first  division 
of  the  limb  coincides  with  zero  of  the  vernier,  the  angle  will  be 
SO'.  In  the  same  manner,  30'  acting  as  a  base,  the  angle  may  be 
increased  to  1°,  and  so  on. 

(  30  ).  Sometimes  verniers  read  lower  than  V,  but  they  are  not 
of  much  practical  use  on  magnetic  instruments.  They  may  also 
differ  in  construction  from  the  kind  described  above,  but  they  all 
work  on  the  same  principle.  The  plate  on  page  26  represents  a 
vernier  compass;  the  vernier  may  be  observed  on  the  compass 
plate  in  front  of  the  box. 

(  31 ).  As  a  general  thing,  when  a  compass  needs  repairing, 
either  from  wear  or  on  account  of  some  mishap,  it  is  best  to  for- 
ward it  to  some  maker  of  mathematical  instruments.  This,  how- 
ever, may  not  always  be  convenient  or  practicable,  and  it  may  be 
well  enough  to  give  some  directions,  which  may  be  of  service  in 
case  of  an  emergency  : 

1.  To  Re-magnetize  the  Xeedk. — When  the  needle  works  lazily, 
on  account  of  losing  a  portion  of  its  magnetism,  it  may  be  re- 
magnetized  with  a  common  bar  or  horse-shoe  magnet  by  passing 
the  south  pole  of  the  magnet  along  the  north  end  of  the  needle 


26  MANUAL   OF  PLANE  SURVEYING. 

from  the  center  to  the  extremity  and  bringing  the  magnet  back  to 
the  starting  point  in  a  circle  of  five  or  six  inches  radius.  The 
south  end  of  the  needle  should  be  treated  in  the  same  manner, 
except  that  the  north  pole  of  the  magnet  should  be  used  on  this 
end.  From  twenty  to  thirty  passes  will  give  it  an  ample  charge. 
2.  To  Sharpen  the  Center-pin. — Sometimes  the  needle  moves  slug- 
ishly  when  the  cenler-pin  upon  which  it  turns  becomes  dull. 


When  this  is  the  case,  take  out  the  plate  in  which  the  center-pin 
is  set  and  then  unscrew  the  pin.  It  may  then  be  sharpened  on  a 
very  fine  stone  and  finished  on  a  piece  of  smooth  leather.  Care 
must  be  taken  to  grind  equally  from  every  side  of  it. 


MANUAL   OF  PLANE  SURVEYING.  27 

3.  To  Replace  a  Spirit-level. —  Eemove  the  brass  tube  from  the 
plate  and  take  off  the  caps  at  the  ends  of  it.     Then  with  some 
pointed  instrument,  as  an  awl  or  a  penknife,  scrape  out  the  plas-  , 
ter  or  other  substance  that  holds  the  vial  in  place,  and  next  force 
out  the  old  vial  by  pressing  on  one  end  of  it.     Xow  slide  the  new 
vial  into  place,  keeping  the  proper  side  up,  and  if  it  is  too  small 
for  the  tube,  wedge  it  up  with  pieces  of  wood  or  paper.     Notice 
carefully  its  position  with  regard  to  the  opening  in  the  tube,  and 
when  it  is  set  in  its  proper  place  press  some  beeswax,  boiled  plas- 
ter, or  putty  of  the  proper  consistency,  around  the  ends  of  it,  so  as 
to  fasten  it  firmly  to  the  sides  of  the  tube;  then  put  on  the  brass 
caps  and  replace  the  tube  on  the  compass  plate.     To  re-adjust  the 
level,  press  on  the  compass  plate  until  the  bubble  stands  in  the 
center  of  the  opening  in  the  tube;  then  turn  the  compass  one-half 
round,  and  if  it  remains  there,  the  level  is  properly  placed,  but  if 
it  runs  toward  the  end  of  the  vial,  and  it  probably  will,  the  end 
toward  which  it  settles  is  too  high,  and  should  be  lowered  or  the 
other  end  raised,  whichever  is  necessary  in  order  to  keep  the  tube 
parallel  with  the  compass  plate.     After  this,  give  the  compass 
another  half-turn  and  repeat  the  process  given  above  until  the 
bubble  will  remain  in  the  middle  of  the  opening  in  the  tube  in 
every  horizontal  position  of  the  plate. 

4.  To  Adjust  a  Ifeiv  Sight. — Fit  it  to  its  place  on  the  plate,  and 
notice  how  the  slit  lines  with  that  of  the  old  one  on  the  opposite 
end.     If  it  inclines  to  one  side,  remove  it  and  file  off  its  base  on 
the  opposite  side  where  it  rests  on  the  plate.    Then  try  it  again,  and 
keep  up  the  operation  until  the  two  slits  coincide  throughout  their 
•whole  length.     If  both  sights  need  adjusting,  hang  a  plumb,  using 
a  fine  thread  or  hair,  and  regulate  both  sights  by  it.     The  com- 
pass should  be  perfectly  level  whenever  an  observation  of   the 
thread  is  taken,  and  the  sights  will  be  properly  adjusted  when- 
ever they  correspond  with  the  plumb-line. 

5.  To  Straighten  the  Center-pin. — Remove  it  with  its  base  from 
the  rest  of  the  compass  and  bend  it  with  a  pair  of  pincers  or 
wrench  made  for  the  purpose,  always  grasping  it  about  an  eighth 
of  an  inch  below  its  point. 

6.  To  Straighten  the  Needle.— It  sometimes  happens  that  the  nee- 
dle of  the  compass  does  not  "  cut "  opposite  degrees  on  the  circle, 
as,  for  instance,  when  its  north  point  is  placed  at  0  its  south  point 
inclines  either  to  the  right  or  left  of  the  opposite  0;  when  this  is 


28  MANUAL   OF   PLANE   SURVEYING. 

the  case  the  error  may  be  corrected  by  bending  the  needle  with  the 
fingers. 

7.  To  put  in  a  new  Glass. — First  take  off  the  brass  ring  that  con- 
tains it  (bezzle  ring)  and  remove  the  putty.     Then  take  out  the 
old  glass  and  put  in  the  new  by  reversing  the  process.     If  the  new 
glass  is  so  large  that  it  will  not  go  in  readily,  hold  the  edge  on  a 
grindstone  and  grind  it  down.     The  manner  in  which  it  should  be 
ground  may  generally  be  seen  by  noticing  the  glass  just  taken  out. 

8.  The  motion  of  the  ball  at  the  head  of  the  Jacob's  staff  may 
be  regulated  by  a  screw-cap  that  fits  down  upon  it.     It  should  be 
kept  reasonably  tight  in  order  that  the  compass  may  not  be  too 
easily  jarred  out  of  level.     If  it  works  loosely,  screw  the  cap  down 
tighter.     After  long  usage  the  ball  may  not  fit  the  cavity  well, 
and  in  this  case  it  may  be  taken  out  and  a  small  piece  of  sheet 
brass  placed  under  it,  or  even  a  piece  of  paper  will  answer  for  a 
short  time. 

(  32  ).  In  carrying  the  compass  it  need  only  be  lifted  from  the 
staff  and  put  under  the  left  arm  so  that  one  of  the  sights  may  pro- 
ject up  behind  the  shoulder,  and  the  staff  makes  a  good  walking 
stick;  but  in  transportation  over  the  country  the  sights  should  be 
taken  off  and  all  packed  snugly  in  a  box  or  something  else  that 
will  answer  the  purpose.  A  suitable  box  is  usually  furnished 
with  the  compass  by  the  manufacturer. 

(  33 ).  All  compasses  are  provided  with  a  lever  or  spring  with 
which  to  raise  the  needle  from  the  center-pin  when  the  compass 
is  not  in  use,  and  this  should  not  be  neglected. 

(  34  ).  The  compass  when  not  in  use  should  be  placed  in  a  hor- 
izontal position  and  the  needle  allowed  to  assume  its  natural  di- 
rection. If  this  precaution  is  taken,  the  needle  will  better  retain 
its  polarity. 

(  35  ).  The  telescope  compass  is  gradually  growing  into  favor 
with  surveyors  and  seems  to  be  taking  the  place  of  the  sight  com- 
pass in  many  localities.  The  telescope  enables  the  surveyor  to- 
set  a  flag  at  longer  ranges,  at  greater  elevations  and  depressions, 
and  discern  it  more  easily  among  trees  and  bushes.  It  is  not  quite 
so  convenient  to  handle  as  the  sight  compass,  however,  but  this  is 
no  great  disadvantage.  Either  may  be  used  on  a  tripod,  instead 
of  a  Jacob's  staff.  For  plate  of  a  telescope  compass  see  frontis- 
piece. This  engraving  represents  the  very  fine  instrument  manu- 
factured by  T.  F.  Randolph,  Cincinnati. 


MANUAL  OF   PLANE  SURVEYING.  29 

QUESTIONS  ON  CHAPTER  II.     " 

1.  Describe  the  compass. 

2.  How  are  the  degrees  numbered  on  the  circumference  of  the 

compass  circle  ? 

3.  How  are  the  sights  arranged  ? 

4.  Describe  the  magnetic  needle.     How  is  the  delicacy  of  a 

needle  determined? 

5.  Explain  the  method  of  measuring  horizontal  angles. 

6.  How  are  vertical  angles  sometimes  approximately  meas- 

ured? 

7.  Why  are  the  letters  E  and  W  reversed  on  the  compass  face? 

8.  State  the  rule  for  taking  the  readings  from  the  compass 

circle. 

9.  The  north  end  of  the  needle  points  20°  to  the  right  of  N. 

What  is  the  bearing  of  the  line  of  sight?  What  is  its 
reverse  bearing? 

10.  What  is  a  northerly  bearing ?     A  southerly  bearing? 

11.  In  running  lines  east  what  letter  on  the  compass  face  should 

be  turned  north?     In  running  west,  what  one?     Why?' 

12.  Give  the  four  rules  for  measuring  angles. 

13.  What  is  the  included  angle  in  each  of  the  following  cases : 

N  40°  E  and  N  62°  E?  S  15°  W  and  S  39°  E?  N  29°  W 
and  S  43°  W  ? 

14.  How  is  electricity  excited  in  the  glass  of  the  compass  ?  How 

may  it  be  removed  ? 

15.  How  do  iron  and  steel  affect  the  action  of  the  needle  ? 

16.  What  two  kinds  of  compasses  are  in  use  ? 

17.  How  many  kinds  of  sight  compasses  are  there  ? 

18.  What  is  the  difference  between  a  plain  compass  and  a  ver- 

nier compass? 

19.  Describe  the  "  vernier."     Of  what  advantage  is  it  ? 

20.  How  is  the  needle  re-magnetized  ? 

21.  Explain  the  manner  in  which  the  center-pin  is  sharpened. 

22.  How  is  a  spirit  level  replaced  ? 

23.  In  what  way  is  a  new  sight  adjusted  ? 

24.  How  do  you  determine,  when  the  two  extremities  of  the 

needle  do  not  "  cut "  opposite  degrees,  whether  it  is  the 
needle  or  the  center-pin  that  is  bent?  Answer — Turn  the 
compass  and  notice  the  amount  of  the  error  in  several 


30  MANUAL  OF  PLANE  SURVEYING. 

positions.  If  it  decreases  in  certain  places  and  increases 
in  others,  it  is  the  center-pin.  If  it  remains  about  the 
same  in  every  position,  it  is  probable  that  the  needle 
alone  is  bent. 

25.  Why  should  the  needle  be  raised  against  the  glass  when 

the  compass  is  not  in  use  ? 

26.  What  is  the  proper  position  for  the  compass  during  the  in- 

terval between  surveys? 

27.  What  are  the  advantages  of  the  telescope  compass  over  the 

sight  compass? 


CHAPTER   III. 

THE  VARIATION  OF  THE  MAGNETIC  NEEDLE. 

(  36  ).  The  meridian  of  any  point  on  the  earth's  surface  is  a  due 
north  and  south  line  connecting  the  point  with  the  poles  of  the 
earth. 

(  37  ).  This  is  called  the  true  meridian  of  the  place,  in  order  to 
distinguish  it  from  the  magnetic  meridian,  which  will  be  considered 
further  on. 

(  38  ).  Various  methods  are  employed  in  determining  the  true 
meridian,  but  only  two  of  the  most  simple  and  satisfactory  will 
be  described : 

1.  By  a  Shadow  Cast  by  a  Perpendicular  Otgect. — Erect  a  perpen- 
dicular staff  on  a  level  surface,  so  that  its  shadow  will  remain  on 


the  surface  from  about  8  o'clock,  A.  M.,  till  4  o'clock,  p.  M.,  as  in- 
dicated in  the  horizontal  projection,  Fig.  3,  in  which  S  N  repre- 
sents the  staff. 

(31) 


32  MANUAL   OF   PLANE  SURVEYING. 

Three  or  four  hours  before  noon,  with  a  radius,  P  S,  shorter 
than  the  length  of  the  shadow,  and  from  the  point  S  as  a  center, 
describe  an  arc  through  the  point  P  and  produce  it  beyond,  opposite 
P.  Then  mark  the  point  P,  where  the  shadow  last  touches  the  arc, 
and  in  the  afternoon  the  other  point  P  where  it  first  touches  it 
again,  and  connect  these  two  points  with  a  line  PP.  Bisect  this 
line  at  O,  and  the  line  SO,  produced  in  either  or  both  directions, 
will  represent  the  true  meridian  of  the  place.  It  is  not  exactly 
correct,  except  at  certain  times  during  the  year  (at  the  solstices), 
but  it  is  always  sufficiently  accurate  for  ordinary  purposes. 

2.  By  the  Polar  Star.— The  Polar  star  (Polaris)  is  situated  about 
1J  degrees  from  the  north  pole  of  the  heavens,  and  appears  to  re- 
volve around  the  pole  once  in  23  hr.  56  min.  If,  now,  we  suppose 
a  vertical  plane  to  pass  through  the  north  pole  and  the  eye  of  the 
observer,  then  twice  during  the  time  of  revolution  Polaris  will  be 
in  this  plane,  and  consequently  in  the  meridian  of  the  observer 
(once  when  above  the  pole  and  again  when  below  it).  These  are 
called  its  upper  and  lower  culminations,  respectively. 

( 1 ).  On  the  opposite  side  of  the  pole  from  Polaris  is  a  star 
known  as  Alioth,  or  more  commonly  as  Epxilcm,  of  the  constellation 
of  the  Great  Bear  or  "  Dipper"  This  is  the  first  star  in  the  handle 
of  the  Dipper,  and  is  situated  next  to  the  four  that  form  the  quad- 
rilateral, the  outside  two  of  which,  Dubhe  and  Merak,  are  called 
"  the  pointers,"  because  they  indicate  the  position  of  Polaris. 

(  2  ).  Since  these  stars,  Polaris  and  Alioth,  are  almost  exactly 
on  opposite  sides  of  the  pole,  it  is  evident  that  they  will  both  be 
on  the  meridian  of  the  observer  when  one  of  them  is  above  the 
other,  and  it  is  more  convenient  to  make  the  observation  during 
the  upper  culmination  of  Polaris. 

(3).  Suspend  a  plumb-line  from  some  elevated  projection,  as 
the  limb  of  a  tree  or  a  strip  nailed  to  the  side  of  a  building  or 
high  post,  and  at  a  point  south,  not  so  distant  that  Polaris  will 
rise  above  the  point  of  suspension  of  the  plumb,  arrange  a  short 
board  horizontally  east  and  west,  a  little  below  the  level  of  the 
eye.  On  this  board  place  some  kind  of  a  contrivance  containing 
an  opening  across  which  a  thread  may  be  stretched  so  that  it  will 
be  parallel  to  the  plumb-line  when  the  instrument  is  in  use,  and 
slide  the  instrument  along  the  board  until  the  thread  ranges  with 


MANUAL   OF   PLANE   SURVEYING. 


the  plumb-line  and  Polaris.  Continue  to  move  the  instrument 
west  as  Polaris  moves  to  its  point  of  superior  culmination,  and 
watch  also  the  approach  of  Alioth  to  the  meridian.  As  soon  as 
the  plumb-line  falls  on  both  stars,  fasten  the  instrument  to  the 


POLARIS. 


NORT.fr 


(-   ALOTH. 


FIG.  4. 


OUBHE. 
WERAK 


board,  and  you  have  two  points  in  the  true  meridian.  The  line 
through  these  may  be  produced  at  pleasure  and  permanently 
marked.  Fig.  4  represents  the  plumb-line  covering  both  stars. 

(4).  Still  greater  accuracy  may  be  reached  by  following  Polaris 
for  twenty-two  minutes  after  the  plumb-line  falls  on  it,  and  then 
marking  the  line. 

(5)-  When  the  upper  culmination  occurs  during  the  day,  the 
lower  culmination  must  be  used,  but  Alioth  is  then  very  high. 

(  39 ).   The  following  table  shows  the  time  of  the  upper  culmin- 
ation of  Polaris  for  each  tenth  day.     The  time  for  intermediate 
days  may  be  approximated  by  interpolation.     The  time  is  given 
to  the  nearest  minute: 
3 


34 


MANUAL   OF   PLANE  SURVEYING. 


MONTH. 

1st  Day. 

llth  Day. 

21st  Day. 

January  .  .  . 

h.  m. 
6  :  30  P.  M. 

h.  m. 

5:  50P.M. 

h.  m. 
4:  50P.M. 

February  .  . 

4  :  27    " 

3  :  47    " 

2  :  48    " 

March  .... 

2  :  32    " 

1:53    " 

12:50    " 

April  
May  

12  :  26    " 
10  :  29  A.M. 

11:  47A.M. 
9  :  49    " 

10:  49  A.M. 

8:  50    " 

June  

8:27    " 

7  :  48    " 

6  :  49    " 

July  

6:29    " 

5:  50    " 

4  :  51    " 

August  .  .  . 

4  :28    " 

3:48    " 

2:49    " 

September  . 

2:  26    " 

1  :  47    " 

12:48    " 

October  .  .  . 

12  :  29    " 

11:  49  P.M. 

10:  50P.M. 

November.  . 

10  :  27  P.  M. 

9:47    " 

8:48    " 

December  .  . 

8:28    " 

7  :  49    " 

6:  50    " 

(40).  If  a  magnetic  compass  be  placed  on  the  true  meridian 
thus  established,  the  needle  of  the  compass,  pointing  toward  thenorth 
magnetic  pole,  instead  of  toward  the  north  poleofthe  earth,  marks  a  line 
called  the  magnetic  meridian,  which  coincides  with  the  true  merid- 
ian in  comparatively  few  places  on  the  earth.  The  angle  formed 
by  the  difference  in  direction  of  these  two  lines  is  called  the  varia- 
tion or  declination  of  the  needle. 

(41).  If  the  north  point  (south  pole)  of  the  needle  points  to  the 
east  of  the  true  meridian,  the  variation  is  said  to  be  east,  and  if  it 
points  to  the  west,  the  variation  is  said  to  be  west.  The  amount  ot 
variation  is  determined  by  the  size  of  the  angle. 

(  42  ).  In  the  United  States  there  is  a  line  extending  southeast 
through  the  eastern  part  of  Michigan,  western  part  of  Lake  Erie, 
eastern  Ohio,  central  West  Virginia,  Virginia,  and  North  Carolina, 
reaching  the  Atlantic  ocean  near  Wilmington,  that  is  called  the 
agmic  line  or  line  of  no  variation,  because  at  all  points  thereon  the 
magnetic  meridian  and  true  meridian  coincide.  Places  east  of 
this  line  have  west  variation,  and  those  west  of  it  have  east  varia- 
tion. 


MANUAL   OF   PLANE   SURVEYING.  35 

( 43  ).  Isogonic  lines  or  lines  of  equal  variation  run  through  places 
having  the  same  variation.  For  instance,  the  line  of  three  degrees 
west  variation  passes  through  Chesapeake  Bay,  Maryland,  central 
Pennsylvania,  and  western  New  York,  and  the  line  of  three  de- 
grees east  variation  passes  through  western  South  Carolina,  eastern 
Georgia,  Tennessee,  Kentucky  and  Indiana,  and  western  North 
Carolina,  Ohio,  and  Michigan.-  The  variation  increases  in  both 
directions  from  the  agonic  line,  reaching  about  18  degrees  west 
variation  in  eastern  Maine  and  22  degrees  east  variation  in  the 
northern  part  of  Washington  Territory. 

(  44 ).  The  isogonic  lines  converge  toward  the  north  magnetic 
pole,  situated  at  present  in  longitude  about  96°  west  from  Green- 
wich, and  latitude  about  70°  north.  This  pole  has  been  gradually 
moving  westward  for  several  years,  and  will  perhaps  continue  to 
do  so  for  several  years  to  come.  This  causes  the  agonic  line  and 
all  the  isogonic  lines  to  move  in  the  same  direction,  so  that  west 
variation  is  constantly  increasing  and  east  variation  constantly 
decreasing  throughout  the  eastern  and  central  portions  of  the 
United  States. 

( 45 ).  The  change  that  takes  place  in  the  variation  of  the 
needle  at  any  place  from  this  cause  is  called  its  secular  change. 
This  varies  in  different  localities,  and  is  generally  greater  in  the 
northern  part  of  the  United  States  than  in  the  southern  part.  It 
is  determined  by  comparing  the  variation  at  the  time  of  any  ob- 
servation with  that  of  a  preceding  or  succeeding  one. 

( 46  ).  But  this  is  not  the  only  change  to  which  the  needle  is 
subject.  Its  action  is  modified  by  other  influences.  The  north 
end  of  the  needle  moves  westward  from  about  6  o'clock  A.  M., 
until  about  2  o'clock  p.  M.,  and  then  gradually  returns  to  the 
starting  point.  This  is  called  its  diurnal  change,  and  it  sometimes 
amounts  to  10  or  12  minutes  of  a  degree. 

This  change  is  about  twice  as  great  in  summer  as  in  winter, 
hence  an  annual  change  must  be  taken  into  consideration. 

( 47  ).  The  following  table,  taken  from  the  Keport  of  the  United 
States  Coast  Survey,  illustrates  these  changes.  The  mean  magnetic 
meridian  is  the  average  position  of  the  needle  for  the  day : 


MANUAL   OF  PLANE  SURVEYING. 


^ 

d 

HOUK. 

A 

1 

a 

a 

S 

i 

£ 

•3 

'S 

.S 

OJ 

"*! 

^ 

/ 

1 

i 

f 

6  A.  M. 

3 

4 

2 

i 

» 

7  "  " 

4  ' 

5 

3 

i 

3  <B 

8  "  " 

4 

5 

3 

2 

I! 

9  "  " 

3 

4 

2 

2 

!« 

10  "  " 

1 

1 

0 

1 

11  "  " 

1 

2 

2 

0 

* 

12  M. 

4 

4 

3 

2 

II 

IP.   M. 

5 

6 

4 

3 

i? 

2  "  " 

5 

5 

3 

3 

^  5" 

of 

3  "  " 

4 

4 

2 

2 

^ 

4  '•  " 

3 

3 

1 

1 

2, 

5  *'  ** 

2 

2 

1 

1 

g 

6  "  " 

1 

1 

0 

0 

3 

( 48 ).  Probably  as  good  a  time  as  any  to  get  the  variation  of 
the  needle  is  between  5  and  7  o'clock  in  the  evening,  as  it  then 
pretty  nearly  indicates  the  mean  magnetic  meridian.  The  diur- 
nal and  annual  changes  are  usually  disregarded  in  practical  work 
with  the  compass. 

(49).  The  needle  is  also  subject  to  disturbances  that  do  not 
appear  conformable  to  any  known  law.  These  generally  take 
place  during  a  thunder  storm,  aurora,  or  other  electrical  phe- 
nomenon, and  sometimes  cause  a  surveyor  no  small  amount  of 
vexation. 

(50).  The  north  end  of  the  needle  is  also  inclined  to  point 
downward.  This  is  called  its  "  dip "  or  inclination,  and  increases 
as  we  go  northward.  It  is  overcome  by  making  the  north  end  of 
the  needle  the  lighter,  and  in  order  to  render  the  needle  adjusta- 
ble, a  small  counterpoise  is  arranged  in  a  groove  on  its  south  end. 

(51 ).  A  great  many  magnetic  phenomena  are  at  present  very 
imperfectly  understood,  and  many  of  the  conclusions  based  upon 


MANUAL   OF   PLANE  SURVEYING.  37 

partially  accepted  theories  are  liable  to  be  overthrown  at  almost 
any  time.     A  discussion  of  these  belongs  to  Philosophy. 

QUESTIONS  ON  CHAPTER  III. 

1.  What  is  meant  by  the  meridian  of  a  place? 

2.  What  is  the  difference  between  the  true  meridian  and  the 

magnetic  meridian? 

3.  Explain  each  of  the  methods  given  for  establishing  a  true 

meridian. 

4.  What  is  meant  by  the  culmination  of  Polaris? 

5.  To  what  does  the  north  end  of  the  needle  point? 

6.  What  do  you  understand  by  "  the  variation  of  the  needle?  " 

7.  What  is  east  variation?     West  variation ? 

8.  What  determines  the  amount  of  variation  ? 

9.  Through  what  part  of  the  United  States  does  the  agonic 

line  run? 

10.  What  are  isogonic  lines? 

11.  What  is  the  situation  of  the  north  magnetic  pole? 

12.  In  what  direction  is  the  agonic  line  and  all  the  isogonic 

lines  moving  at  present? 

13.  What  effect  does  this  movement  have  on  east  variation? 

West  variation? 

14.  What  other  influences  act  upon  the  needle? 

15.  What  is  the  difference  between  the  diurnal  change  and  the 

annual  change? 

16.  What  is  meant  by  "dip?" 

17.  How  is  it  overcome  ? 


CHAPTER    IV. 

EFFECT  OF  CHANGE  OF  VARIATION*  ON  OLD  LINES  AND  METH- 
ODS OF  CORRECTING  BEARINGS. 


FlG.  5. 


<!The  terms  variation  and  declination  have  become  synonymous  in  mean- 
ing, although  it  would  be  better,  etymologically  speaking,  to  use  the  word 
variation  to  denote  the  change  of  declination.  We  would  then 
of  declination,  instead  of  change  of  variation  as  at  present. 

(38) 


MANUAL   OF   PLANE  SURVEYING. 


30 


(  52  ).  As  the  variation  of  the  needle  is  constantly  changing,  so 
the  bearing  of  a  line  surveyed  at  any  time  is  also  subject  to  con- 
stant change.  This  renders  it  necessary  in  the  re-survey  of  any 
line  to  take  into  consideration  the  amount  of  change  that  has  taken 
place  since  the  previous  survey;  otherwise  the  location  of  the  line 
will  be  changed. 

To  illustrate :  Let  the  line  A  B,  Fig.  5,  represent  the  line  of 
direction  of  the  magnetic  needle  (magnetic  meridian)  at  a  certain 
time,  and  the  line  C  D  which  makes  an  angle  of  22°  with  the  mag- 
netic meridian,  and  whose  course  is  N  22°  E,  represent  a  line  sur- 
veyed while  the  needle  marks  this  meridian. 

Since  the  north  end  of  the  needle  is  continually  moving  toward 
the  west,  it  is  evident  that  the  angle  between  A  and  C,  formed  by 
the  crossing  of  the  two  lines,  will  be  constantly  growing  larger, 
and  that  the  bearing  of  the  line  C  D  will  consequently  increase 

f         C 


Fro.  6. 


40 


MANUAL   OF   PLANE  SURVEYING. 


with  each  succeeding  year,  as  long  as  the  westward  movement  of 
the  needle  continues. 

(  53  ).  Let  us  now  suppose  that  several  years  after  this  survey 
is  made,  the  needle  assumes  the  position  of  the  dotted  line  E  B, 
Fig.  6,  making  an  angle  with  A  B  equal  to  5°.  The  line  C  D  will 
now  make  an  angle  with  the  magnetic  meridian  equal  to  (22°  -j- 
5°)  =  27°.  It  will  be  seen  that  the  bearing  of  the  line  has  in- 
creased 5°,  the  change  of  variation  of  the  magnetic  needle. 

(  54  ).  If  now  the  surveyor,  through  ignorance  or  carelessness, 
should  neglect  to  take  this  change  into  consideration,  and  run  the 
line  at  the  old  bearing  (22°)  marked  in  the  description  of  the  line, 
he  would  change  its  position  to  that  of  the  dotted  line  D  F,  and 
it  is  easy  to  see  what  an  error  this  would  cause  in  the  survey  of 
a  tract  of  land. 

(  55 ).  A  survey  of  this  kind,  however,  does  not  affect  the  form 
or  area  of  a  piece  of  land,  but  simply  changes  its  boundaries  and 
seems  partly  to  revolve  the  tract  about  the  corner  from  which 
the  surveyor  starts,  as  a  center.  For  instance,  suppose  a  mistake 


==r- -.B 


FIG.  7. 

of  this  kind  to  be  made  in  the  survey  of  the  tract  A  B  C  D,  Fig. 
7 ;  its  position  would  be  changed  to  that  of  the  dotted  quadri- 
lateral. In  this  survey  the  corner  D  was  made  the  starting  point. 
( 56 ).  In  order  to  determine  the  present  bearing  of  a  line,  it  is 
an  advantage  to  know  three  things :  1.  The  bearing  of  the  line 


MANUAL   OF  PLANE  SURVEYING.  41 

at  the  time  of  a  previous  survey.  2.  The  riumber  of  years  that 
have  elapsed  since  that  survey  was  made.  3.  The  annual  amount 
of  secula'r  change  of  variation. 

1  and  2.  Where  field-notes  of  the  former  survey  have  been  pre- 
served, they  generally  and  should  always,  give  the  necessary  in- 
formation in  regard  to  the  bearing  of  the  line  and  date  of  the  sur- 
vey ;  but  where  no  record  of  the  survey  can  be  found,  the  bearing 
may  sometimes  be  approximately  determined  from  old  deeds  or 
descriptions  of  the  land,  and  these  frequently  assist  the  surveyor 
in  arriving  at  a  conclusion  with  regard  to  the  time  the  survey 
was  made.  Persons  living  in  the  neighborhood  may  also  know 
something  of  the  previous  survey. 

3.  In  order  to  determine  the  annual  change  of  variation,  vari- 
ous methods  may  be  employed. 

(1).  The  bearing  of  a  line  at  the  present  time  may  be  compared 
with  that  recorded  by  a  competent  surveyor  some  time  previous. 

(2).  A  true  meridian  may  be  established  by  the  second  method 
explained  in  the  preceding  chapter,  and  the  variation  of  the  needle 
found  from  this  by  observing  the  angle  the  magnetic  meridian 
makes  with  the  true  meridian.  This  result  may  then  be  com- 
pared with  the  variation  recorded  after  a  previous  observation. 

In  both  of  these  cases  the  amount  of  change  in  minutes  will 
equal  the  quotient  obtained  by  dividing  the  difference  of  varia- 
tion in  minutes  between  the  two  observations  by  the  number  of 
years  that  have  elapsed  between  them.  For  instance,  if  the  varia- 
tion of  the  needle  at  a  certain  time  is  found  to  be  5°,  and  in  twenty 
years  after  an  observation  shows  that  it  has  decreased  to  4°,  the 
annual  change  will  equal  (5°  —  4°)  =  1°=  60',  which  divided  by 
20  =  3'.  From  this  the  variation  for  any  subsequent  time  may 
be  found  by  multiplying  the  annual  change,  3',  by  the  number  of 
years,  and  adding  or  subtracting  the  product,  as  will  be  explained 
further  on. 

(3).  When  the  annual  change  at  several  important  points  is 
known,  the  change  at  intermediate  points  throughout  the  country 
may  be  determined  by  interpolation,  but  this  method  is  not  trust- 
worthy in  all  cases. 

(57).  The  following  table  is  taken,  somewhat  modified,  from 
the  report  of  the  U.  S.  Coast  Survey  of  a  few  years  ago,  and  con- 


42  MANUAL   OF   PLANE   SURVEYING. 

tains   the  variation  of   the  needle   at   various   important  stations 
throughout  the  United  States  for  the  year  1881 : 


Bangor,  Me 16°  Sff  W. 

Wolfsboro,  N.  H 12°  15'  W. 

Middlebury,  Vt 11°  35*  W. 

Marblehead,  Mass 1?°  54'  W. 

Schenectady,  N.  Y., 9-  58'  W. 

Chambersburg,  Pa 4°  33'  W. 

New  Haven,  Conn 9°  Off  W. 

Providence,  R.  I., 11°  27'  W. 

Wilmington,  N.  C., 1°  04'  W. 

Jackson,  Miss.,      . 6°  33'    E. 

Washington,  D.  C 3°  17'  W. 

Camden,  N.  J 6°  W  W. 

Columbus,  O., 0°  W  E. 

Savannah,  Ga 1°  50*   E. 

New  Orleans,  La 6°  33'  E. 

Quincy.Wis 6°  33'    E. 

Decatur,  Ind., 1°  5ff   E. 

Grand  Rapids,  Mich 1°  5(X   E. 

Havana,  Cuba 4°  46'   E. 

Carlinville,  111 6°  33'   E. 

Aiken,  8.  C 1°  50   E. 

Monterey,  Cal 16°  06'   E. 

Sitka,  Alaska ...  28°  27'    E. 


This  list  is  not  full  enough  to  be  of  much  practical  use,  but  I 
think  best  not  to  extend  it  any  further,  as  the  surveyor  can  easily 
find  the  variation  at  any  point  himself. 

( 58 ).  The  next  table,  also  taken  from  the  Eeport  of  the  Super- 
intendent of  the  Coast  Survey,  gives  the  annual  amount  of  secular 
change  in  certain  localities  in  the  United  States. 
Maine,  Long  Island,   Delaware,   Maryland, 
Virginia,  South  Florida,  South  Alabama 

and  Mississippi,  and  New  Jersey 3' 

East  Maine,  West  Tennessee,  Missouri,  and 
West  Louisiana 2' 


MANUAL  OF  PLANE  SURVEYING.  •      4 

Ohio,  East  Tennessee,  East  Louisiana,  and 

East  Massachusetts 2'  15"  to  2'  45" 

Kew  Hampshire,  West  Massachusetts,  Khode 

Island,  Connecticut,  AVest  Virginia,  North 

Carolina,   South   Carolina,   Georgia,   and 

North  Florida 3'  15"  to  3'  45" 

North  and  West  New  York 4'  30" 

Vermont 5' 30" 


In  all  these  places  the  change  marked  to  the  right  indicates  an 
increase  of  westerly  or  decrease  of  easterly  variation. 

(  59  ).  These  tables  will  enable  us  to  approximate  to  the  vari- 
ation of  the  needle  at  points  included  in  both  tables,  for  perhaps 
several  years.  For  example,  the  variation  of  the  needle  at  Cam- 
den.  N.  J.,  this  year  is  6°  50'  W;  in  1885  it  will  be  about  6°  50/+ 
(4  X  3')  =  7°  02'.  If  the  variation  is  easterly,  as  in  the  western 
part  of  the  United  States,  the  amount  of  change  is  subtractive, 
instead  of  additive.  Take  this  for  example:  What  will  be  the 
variation  of  the  needle  at  New  Orleans  in  1884? 

1884—1881=3  years. 

Variation  in  1881  =  6°  33'. 

Annual  change  about  2'  30". 

Amount  of  change  =  (3  X  2'  30")=7'  30". 

Variation  in  1 884=6°  33'— 7/30"=6°  25'  30". 

(60).  In  order  to  find  what  the  variation  of  the  needle  was  a 
few  years  ago,  we  have  only  to  reverse  the  rule  given ;  that  is,  add 
where  the  variation  is  easterly,  and  subtract  where  it  is  westerly. 

The  following  examples  may  be  used  for  practice: 

( 1 ).  The  variation  of  the  needle  at  a  certain  place  in  1880  was 
N  5°  32'  E,  and  the  annual  change  2'  20" ;  what  will  be  the  vari- 
ation 1889? 

(2).  In  1878  the  variation  at  a  certain  place  was  4°  15"  W, 
annual  change  3'  30";  what  will  be  the  variation  in  1886? 

(3).  In  1880  the  variation  at  A  was  2°  36'  W;  what  was  the 
variation  in  1874,  supposing  the  annual  change  to  be  4'? 

( 4 ).  The  annual  change  at  B  is  3'  45"  and  the  variation  of  the 
needle  in  1879,  3°  47'  E;  what  was  the  variation  at  this  place  in 
1870? 


44 


MANUAL   OF   PLANE  SURVEYING. 


( 61  ).  It  would  frequently  be  better  if  the  bearings  of  lines 
were  taken  from  the  true  meridian  as  a  basis,  as  no  change  would 
then  take  place  in  the  bearings. 

This  may  be  easily  explained.     Suppose  the  line  A  B  to  repre- 


A      C 


FIG.  8. 

sent  a  section  of  a  true  meridian  of  the  earth,  and  the  line  C  D  to 
represent  one  of  the  boundaries  of  a  piece  of  land.  Both  of  these 
lines  are  fixed,  and  consequently  the  angle  between  them  can 
never  change. 

(  62 ).  If,  however,  the  line  A  B  represents  a  section  of  the  mag- 
netic meridian,  it  is  constantly  changing,  and  this  of  course  affects 
the  angle. 

(  63  ).  In  order  to  determine  the  true  bearing  of  a  line  from  its 
magnetic  bearing,  two  cases  must  be  considered  : 

1.  When  the  variation  is  west.  RULE.— Add  the  variation  to  the 
bearing  for  northwest  and  southeast  courses,  and  take  their  differ- 
ence for  northeast  and  southwest  courses. 

In  Fig.  9  let  A  B  represent  a  due  north  and  south  line  and  C  B 
represent  the  direction  of  the  needle.  D  B  and  E  B  are  two  lines 
whose  bearings  are  to  be  changed  from  the  magnetic  bearing  to 
the  true  bearing.  Suppose  the  variation  of  the  needle  to  be  10° 
W ;  then  the  true  bearing  of  the  line  E  B  will  be  N  (25°  +  10°) 


MANUAL   OF   PLANK  SURVEYING. 


45 


W  =  N  35°  W,  and  the  true  bearing  of  the  line  B  D  will  be  N 
(35°  —  10°)  E  =  N  25°  E. 


FIG.  9. 

2.  When  the  variation  is  east.  RULE. — Reverse  the  preceding  op- 
eration ;  that  is,  take  the  difference  between  the  bearing  and  vari- 
ation for  northwest  and  southeast  courses,  and  their  sum  for  north- 
east and  southwest  courses. 

The  line  A  B,  Fig.  10,  represents  a  section  of  the  true  meridian, 
and  the  line  B  C  a  section  of  the  magnetic  meridian.  As  before, 
the  bearings  of  the  lines  D  B  and  E  B  are  to  be  changed  to  the 
true  bearings  from  the  magnetic  bearings.  If  the  variation  of  the 
needle  be  10°  E,  the  true  bearing  of  the  line  B  D  will  be  N  (25° 
-f  10°)  E  =  N  35°  E,  and  the  true  bearing  of  the  line  B  E  will  be 
N  (35°  —  10°)  W  =  N  25°  W. 


MANUAL,  OF  PLANE  SURVEYING. 


(64).  The  variation  in  the  following  examples  is  5°E;  what  is 
the  true  bearing  of  each  course  ? 

1.  N  45°  W. 

2.  N  32°  W. 

3.  S  16°  W. 

4.  N  17°   E. 

5.  S  25°   E. 

6.  N  25°   E. 

(65).  If  in  any  case  the  sum  of  the  variation  and  bearing 
amounts  to  more  than  90°,  the  supplement*  of  the  sum  must  be; 
taken,  and  the  first  letter  changed  to  its  opposite. 

Take,  for  instance,  the  bearing  N  88°  W  when  the  variation  is 
6°  W.  This  gives  us  the  following  result :  88°  +  6°  =  94°,  and 
taking  94°  from  180°  to  get  the  supplement,  we  have  (180°  —  94°) 

*The  supplement  of  an  arc  is  what  remains  after  subtracting  the  arc  from 


MANUAL   OF  PLANE   SURVEYING.  47 

=  86°.     After  changing  the  first  letter  of  the  magnetic  bearing, 
we  see  that  the  true  bearing  of  the  line  is  S  86°  W. 

(  66  ).   In  changing  from  the  true  to  the  magnetic  bearing,  it  is 
necessary  only  to  reverse  the  foregoing  rules. 

QUESTIONS  ON  CHAPTER  IV. 

1.  What  effect  does  the  change  of  variation  of  the  magnetic 

needle  have  upon  the  bearings  of  lines? 

2.  Why  must  this  change  be  taken  into  consideration  in  a  re- 

survey  ? 

3.  Why  would  it  not  affect  the  figure  or  contents  of  a  tract  of 

land  if  this  change  were  not  regarded  ? 

4.  What  three  things  are  necessary  in  order  to  determine  the 

present  bearing  of  a  line  from  its  magnetic  bearing  at  a 
former  time,  without  a  re-survey  of  the  line? 

5.  How  is  its  bearing  at  the  time  of  a  former  survey  found  ? 

The  annual  change  ? 

6.  Two  observations  thirty  years  apart  show  a  change  of  1°  15' 

in  the  variation  of  the  needle.    What  is  the  annual  change? 

7.  What  is  meant  by  east  variation?     West  variation. 

8.  How  do  you  find  the  true  bearing  of  a  line  from  the  mag- 

netic bearing  ?    The  magnetic  from  the  true  ? 


CHAPTER  V. 

METHOD  OF  RUNNING  LINES,  ETC. 

( 67 ).  The  first  thing  a  surveyor  does  in  making  a  survey  is  to 
find  a  corner  from  which  to  start,  and  this  is  frequently  a  matter 
of  no  little  trouble,  but  its  discussion  belongs  further  on  in  our 
work.  Let  us  suppose  for  the  present  that  the  monument  which 
marks  the  corner  (generally  a  stone  or  a  stake)  is  still  standing,  or 
that  the  witnesses  to  it  are  yet  to  be  found.  These  witnesses  are 
generally  trees,  although  any  other  immovable  object  will  answer 
the  purpose  equally  well.  When  the  corner  is  put  down  the  course 
and  distance  from  it  to  one  or  more  of  these  objects  are  carefully 
noted  in  the  record  (field  notes)  of  the  survey,  so  that,  no  matter 
if  the  monument  that  marks  the  corner  has  disappeared,  its  posi- 
tion can  easily  be  determined  as  long  as  any  of  the  witnesses 
remain.  The  witnesses,  if  trees,  are  called  witness- trees,  and 
are  marked  about  eighteen  inches  from  the  ground  on  the  side 
facing  the  corner.  The  mark  consists  of  a  blaze  about  six  or  eight 
inches  long,  which  generally  has  a  cross  notch  cut  in  it. 

( 68 ).  If  the  monument  at  the  corner  remains,  its  center  should 
be  taken  as  the  point  from  which  to  run  the  line;  but  if  it  can  not 
be  found,  its  position  must  be  determined  from  the  witnesses.  It 
is  probable  that  the  approximate  location  of  the  corner  is  known, 
so  that  the  surveyor  or  some  of  his  attendants  know  where  to  look 
for  it.  Suppose  the  surveyor  examines  his  record  and  finds  that 
when  the  corner  was  set,  a  beech  tree  18  inches  in  diameter  was 
taken  as  a  witness  to  it.  This  tree  stood  in  a  direction  N  43°  E 
from  the  corner,  and  was  distant  78  links.  He  looks  in  that  direc- 
tion and  sees  a  tree  which  he  believes  to  be  the  one,  and  after  an 
examination  he  finds  the  surveyor's  mark  and  is  convinced  of  its 
identity.  All  he  has  to  do  now  is  to  set  the  compass  in  front  of  the 
(48) 


?.r;6 


MANUAL   OF   PLANE  SURVEYING.  49 

mark  on  the  tree  and  turn  it  until  the  needle  indicates  the  reverse 
bearing,  S  43°  W;  a  staff  is  then  set  in  the  line  of  sight  somewhat 
further  from  the  tree  than  the  distance  given  for  the  corner,  and  a 
measurement  of  78  links  made  from  the  notch  on  the  tree  in  the 
direction  of  the  staff;  the  termination  of  this  measurement  is  the 
corner. 

( 69 ).  On  prairies,  where  no  trees  are  available  for  witnesses, 
mounds  of  earth  thrown  up  around  a  stake  are  usually  made  to 
answer  the  purpose,  and  in  clearings  it  is  sometimes  necessary  to 
make  the  same  kind  of  witnesses,  or  supply  their  places  with  stones 
set  at  suitable  distances  from  the  corner.  A  good  monument 
should  always  be  put  down  at  the  corner,  and  one  not  easily 
moved  or  destroyed  does  not  particularly  need  any  witnesses  to  it. 

(70).  The  method  of  describing  a  witness-tree  employed  by 
surveyors  in  making  up  their  field-notes,  is  much  abbreviated  from 
the  method  used  above.  The  above  description  would  read  as  fol- 
lows :  Be  18  N  43  E  78.  The  first  number  signifies  the  diameter, 
the  second  the  bearing,  and  the  third  the  distance  in  links.  The 
bearing  and  distance  of  the  tree  from  the  corner  are  called  its 
"course  and  distance."  Likewise  the  course  and  distance  of  any 
line  are  its  bearing  and  length. 

(  71 ).  After  the  corner  has  been  found,  the  next  thing  is  to  run 
a  line  from  this  corner  to  the  next.  In  surveying,  a  line  always 
connects  two  corners,  and  in  plane  surveying  it  represents  the 
shortest  distance  between  them. 

(  72  ).  In  order  to  run  this  line,  its  course  and  distance  must  be 
at  least  approximately  known,  and  should  be  exactly  known. 

(  73  ).  Let  us  suppose  that  the  true  bearing  of  the  line  is  N  3°  E, 
and  its  length  6  chains  and  50  links.  Since  the  true  bearing  ia 
given  it  has  remained  unchanged  (Art.  61),  and  the  surveyor  has 
only  to  turn  the  vernier  of  the  compass  until  he  sets  off  the  varia- 
tion of  the  needle,  and  then  set  the  north  end  of  the  needle  at  N 
3°  E.  The  sights  of  the  compass  are  then  set  Oil  the  line,  and  a 
measurement  of  6  chains  and  50  links  on  the  line  of  sight  will 
bring  him  to  the  next  corner. 

(  74).  In  setting  off  the  variation  of  the  needle  on  the  vernier, 
two  cases  arise:  (1)  when  the  variation  is  east,  and  (2)  when  the 
variation  is  west. 

1.  To  Set  Of  East  Variation.  KULE.— Turn  the  sights  so  as  to 
throw  the  line  of  sight  to  the  left  until  the  angle  between  the  line 
4 


50  MANUAL  OF  PLANE  SURVEYING. 

of  sight  and  the  0  points  of  the  compass  circle  equals  the  vai-i- 
ation  of  the  needle. 

2.  To  Set  Off  West  Variation.  RULE.— Turn  the  sights  of  the  com- 
pass so  as  to  throw  the  line  of  sight  to  the  right  until  the  angle  be- 
tween the  line  of  sight  and  the  0  points  of  the  circle  equals  the 
variation. 

These  rules  obtain  whether  the  movement  of  the  needle  is  toward 
the  west  or  east,  and  the  vernier  should  generally  not  be  changed 
during  the  survey. 

We  are  now  prepared  to  run  lines  whose  true  bearings  are 
known. 

The  course  and  distance  of  a  line  are  N  43°  E 16  chains  and  50  links. 

(  76 ).  After  finding  the  first  comer  on  this  line,  the  surveyor 
sets  off  the  variation  of  the  needle  on  the  vernier,  and  then  sets 
the  compass  at  the  corner,  levels  it,  and  turns  the  sights  until  the 
needle  indicates  the  bearing  of  the  line.  He  then  starts  the  flag- 
man in  the  direction  of  the  line  of  sight  to  a  suitable  distance 
from  the  compass,  and  then  moves  him  either  to  the  right  or  left 
until  the  sights  strike  the  flag-staff.  The  staff  is  left  in  this  posi- 
tion until  the  surveyor  comes  up  and  sets  the  compass  where  it 
stood,  and  then  the  flagman  seeks  another  position  further  along 
the  line. 

(  76  ).  The  chain-men  commence  at  the  center  of  the  monument 
that  marks  the  corner  as  soon  as  the  surveyor  has  set  the  flag  on 
the  line.  The  one  that  takes  the  lead  is  called  the  "  leader,"  and 
the  one  at  the  rear  end  of  the  chain  is  called  the  "  follower."  At 
setting  out,  the  leader  takes  a  certain  number  of  iron  or  steel  pins 
(usually  ten)  and  puts  one  down  at  the  end  of  the  chain.  The 
chain  is  then  carried  forward  another  length  and  its  rear  end  held 
against  the  pin  just  put  down  while  the  leader  sticks  another  into 
the  ground.  Great  care  should  be  taken  to  keep  the  chain  free  of 
"  kinks,"  taut,  and  as  nearly  horizontal  as  possible,  no  matter  how 
uneven  the  ground  may  be,  and  the  follower  should  line  each  pin 
with  the  flag-staff  or  compass,  whichever  may  be  set  at  the  point 
to  which  they  are  running. 

(77)  When  ten  pins  are  nsed  the  "marker"  drives  a  stake* 
where  the  last  pin  stood ;  the  leader  takes  the  ten  pins  from  the 
follower,  and  the  measurement  proceeds  just  the  same  as  from  the 
corner.  If  eleven  are  used  the  follower  retains  one  in  starting  out 
from  the  corner,  and  as  soon  as  the  leader  puts  down  the  last  of 

*The  points  on  the  line  marked  by  the  stakes  are  called  "outs,"  because 
the  leader  runs  out  of  marking  pins  at  these  places. 


MANUAL  OF   PLANE   SURVEYING.  51 

his  ten  he  receives  ten  from  the  follower,  and  the  new  measure- 
ment begins  at  the  last  pin  put  down,  instead  of  at  the  stake,  as 
hefore.  As  we  said  before,  the  chain  usually  employed  in  survey- 
ing is  only  two  rods,  fifty  links,  or  thirty-three  feet  in  length,  and 
since  a  stake  is  put  down  at  every  ten  lengths,  it  leaves  the  stakes 
twenty  rods,  or  five  chains  apart.  The  first  stake  from  the  corner 
has  one  notch  cut  in  it,  the  second  two,  the  third  three,  and  so  on. 
The  surveyor  should  sight  back  while  the  marker  is  setting  the 
stake  and  see  that  it  is  put  exactly  on  the  line. 

(  78 ).  The  relative  places  on  the  line  of  the  persons  engaged  in 
the  survey  are  as  follows :  (1)  flagman,  (2)  surveyor,  (3)  chain 
carriers  and  marker. 

(  79 ).  The  measurement  is  continued  in  the  direction  N  43°  E 
until  the  length  of  the  line,  16  chains  and  50  links,  has  been  meas- 
ured, and  if  the  line  thus  run  strikes  the  corner,  it  coincides  with 
the  true  line,  but  if  it  does  not,  the  stakes  must  be  moved,  either 
to  one  side  or  the  other,  until  it  is  made  to  coincide. 

(  80).  Let  us  suppose  that  in  the  case  before  us  the  line  has 
been  found  to  terminate  at  a  point  8^  links  to  the  left  of  the  cor- 
ner. This  line  is  called  a  "  random  line,"  and  it  is  evident  that 
it  has  been  diverging  from  the  true  line  gradually  since  leaving 
the  starting  point,  and  that  this  divergence  is  always  in  propor- 
tion to  the  distance  from  this  point. 

Dividing  now  the  distance  between  the  terminations  of  the  two 
lines  by  their  length  in  chains,  we  have  8}  -5-  16|  =  £  link  for 
the  increase  of  divergence  for  each  chain  from  the  point  from 
which  the  lines  start,  and  since  the  first  stake  is  5  chains  from 
this  point,  it  is  5  X  2  link  =  2£  links  off  the  true  line.  In  like 
manner  the  second  stake  is  10  X  £  link  =  5  links  off  the  true 
line,  and  the  third  one  15  X  J  link  =  7\  links  off  the  true  line. 
The  stakes  are  put  on  the  true  line  by  moving  them  the  given  dis- 
tances in  the  proper  direction  which  is  always  the  opposite  of  the 
direction  of  the  termination  of  the  random  line  from  the  corner. 

(  81 ).   This  may  be  illustrated  by  the  figure. 


FIG.  11. 


52  MANUAL   OF   PLANE   SURVEYING. 

Let  A  C  represent  the  true  line,  and  A  B  the  random  line. 
Since  the  termination  of  the  random  line  lies  to  the  left  of  the  true 
corner,  the  stakes  must  be  moved  to  the  right.  The  numbers  on 
the  random  line  indicate  the  order  in  which  the  stakes  are  num- 
bered, and  those  on  the  true  line  represent  the  distance  that  each 
stake  must  be  moved  to  the  right. 

(  82  ).  This  enables  us  to  deduce  the  following  rule  for  correcting 
the  stakes:  Divide  the  distance  between  the  terminations  of  the 
two  lines  in  links  by  their  common  length  in  chains,  and  multi- 
ply the  quotient  by  the  number  of  chains  the  stake  is  distant  from 
the  starting  point.  The  product  will  equal  the  number  of  links 
the  stake  is  to  be  moved.  Then  move  the  stake  in  a  direction  op- 
posite to  that  of  the  termination  of  the  random  line  from  the 
corner. 

Where  the  length  of  the  line  is  a  multiple  of  five  chains  (the 
distance  between  stakes),  the  following  rule  will  be  found  more 
simple :  Divide  the  distance  between  the  terminations  of  the  two 
lines  in  links  by  the  number  of  stakes  on  the  line,  and  multiply 
the  quotient  by  the  number  of  the  stake  from  the  starting  point. 
The  product  indicates  the  number  of  links  the  stake  is  to  be  moved 
as  before. 

(  83  ).  In  both  of  these  rules  the  terminations  of  the  two  lines 
and  the  starting  point  are  considered  as  the  vertices  of  an  isosceles 
triangle,  and  the  distance  between  the  terminations  of  the  two 
lines  should  be  measured  on  a  line  that  will  make  the  same  angle 
with  one  as  with  the  other. 

(  84  ).  In  the  following  examples,  give  the  distance  and  direc- 
tion each  stake  should  be  moved  : 

(1).  Length  of  line,  20  chains.  Ran  to  the  left  of  true  corner 
40  links. 

(2).   Line  40  chains  long.     Ran  to  the  left  20  links. 

(3).   Line  20  chains.     Ran  to  the  right  30  links. 

(4).   Line  18  chains.     Ran  to  the  left  12  links. 

(5).   Line  16  chains.     Ran  to  the  right  32  links. 

(6).   Line  26  chains.     Ran  to  the  right  45i  links. 

(  85  ).  In  making  up  his  field-notes,  the  surveyor  usually  writes 
links  as  hundredths  of  a  chain.  For  instance,  ten  chains  and 
forty  links  would  be  written  10.40.  He  also  sometimes  uses  the 
word  "  missed  "  to  denote  that  the  termination  of  the  random  line 
lies  either  to  the  right  or  left  of  the  true  corner,  and  accompanies 


MANUAL,   OF   PLANE   SURVEYING.  53 

it  with  the  word  "  right "  or  "  left,"  as  the  case  may  be.  These 
terms  are  frequently  abbreviated  in  writing  by  using  their  initial 
letters  instead  of  the  words  themselves.  Thus,  "  missed  left "  may 
be  cut  down  to  M.  L. 

(  86  ).  Owing  to  the  imperfection  of  magnetic  instruments  it  is 
common,  particularly  on  long  lines,  for  a  surveyor  to  run  either 
to  one  side  or  the  other  of  the  corner  to  which  he  is  running,  even 
when  he  knows  the  bearing  of  the  line.  In  this  case,  however,  he 
seldom  misses  the  corner  any  considerable  distance — perhaps  only 
two  or  three  links — and  the  bearing  he  has  taken  may  be  assumed 
to  be  correct,  the  error  being  due  to  the  manipulation  of  the  in- 
strument. 

(  87  ).  But  where  the  bearing  is  only  approximately  known,  the 
random  line  may  vary  greatly  from  the  true  line,  and  when  this 
happens  the  assumed  bearing  must  be  corrected  in  order  that  the 
true  bearing  may  be  subsequently  determined  without  a  resurvey 
of  the  line. 

(  88 ).  Suppose  a  line  40  chains  long  is  to  be  surveyed,  and 
from  the  best  evidence  the  surveyor  has,  he  assumes  its  bearing  to 
be  N  2°  E.  The  line  is  accordingly  surveyed,  and  upon  reaching 
the  other  end  the  surveyor  finds  that  he  has  missed  the  corner  70 
links.  It  is  plain  that  he  assumed  a  bearing  considerably  in 
error,  and  he  wishes  to  correct  it.  To  do  this,  he  may  employ 
any  of  the  following  rules: 

1.  Multiply  the  length  of  the  line  by  .01745  and  divide  the  pro- 
duct into  the  product  of  the  distance  between  the  extremities  of 
the  two  lines  by  GO.  The  quotient  will  be  the  correction  in  min- 
utes of  a  degree.  In  this  case  we  have 

(.70X60)-r-(.0174nX40)=60/+=l°+. 

All  distances  should  be  in  chains  and  hundredths  of  a  chain. 

This  rule  is  derived  as  follows :  In  Fig.  12  the  lines  A  B  and  A 
C  represent  the  true  line  and  random  line,  respectively,  of  the 
survey.  C  B  represents  the  distance  between  their  extremities. 
'  (1).  The  line  CD  is  called  the  sine  of  the  angle  B  AC,  and  where 
the  angle  is  small,  it  does  not  vary  much  in  length  from  the  chord 
of  C  B  which  connects  the  extremities  of  the  two  lines.  The  line 
A  D  is  called  the  cosine  of  the  angle  B  A  C,  and  where  the  angle 
is  small  it  does  not  differ  materially  in  length  from  the  line  A  B. 

(2).   The  sine  of  an  angle,  therefore,  is  a  perpendicular  raised 


54  MANUAL  OF  PLANE  SURVEYING. 


GL  i    " 

FIG.  12. 

from  one  of  its  sides  to  the  other,  and  the  cosine  of  an  angle  is  the 
portion  of  one  of  its  sides  intercepted  between  the  foot  of  the  sine 
and  the  vertex  of  the  angle  itself.  In  these  definitions  no  distinct- 
ion is  made  between  the  sine  or  cosine  of  the  angle  and  the 
sine  or  cosine  of  the  arc. 

(3).  Particular  attention  should  be  paid  to  the  relation  of  the 
sine  and  cosine  to  one  another,  and  they  should  also  be  studied  in 
their  application  to  the  lines  of  the  survey,  as  explained  above. 
The  surveyor  needs  to  employ  sines  and  cosines  frequently. 

(4).  If  now  we  call  the  length  of  the  radius  1,  the  length  of  the 
sine  of  an  angle  of  one  degree  will  be  .01745  which  is  the  number 
we  employed  in  the  rule.  The  sine  will  increase  in  the  same  ratio 
that  the  radius  increases.  Therefore,  if  the  radius  is  40,  the  sine 
will  be  (40  X  -01745)  =  .69800;  and  as  this  is  to  the  distance  be- 
tween the  extremities  of  two  lines,  so  is  60,  the  number  of  minutes 
in  a  degree,  to  the  number  of  minutes  of  correction ;  hence  the 
rule. 

(5).  Where  the  angle  is  large,  .the  sine  may  differ  considerably 
in  length  from  the  chord,  but  this  is  a  case  that  will  seldom  come 
up  in  practice,  except  where  the  surveyor  chances  to  make  a  mis- 
take, and  in  this  case  he  would  better  resurvey  the  line. 

2.  In  rectangular  surveying,  nearly  all  the  lines  to  be  surveyed 
in  usual  practice  in  dividing  and  sub-dividing  the  section,  are  i 
mile,  $  mile,  or  one  mile  in  length.  Now,  the  sine  of  an  angle  of  one 
degree  for  a  radius  of  20  chains,  or  J  mile,  is  very  nearly  35  links ; 
therefore,  for  40  chains,  or  \  mile,  it  is  (2  X  35)  =  70  links;  and 
for  80  chains,  or  1  mile  it  is  (4  X  35)  =  140  links. 

(1).   This  enables  us  to  modify  the  rule  already  given,  as  follows : 


MANUAL   OF   PLANE  SURVEYING.  55 

Multiply  the  distance  between  the  extremities  of  the  two  lines  by 
60,  and  divide  the  product  by  the  sine  of  one  degree  for  a  radius 
equal  to  the  length  of  the  line  surveyed.  The  quotient  will  be 
the  number  of  minutes  of  correction  to  be  made. 

(2).  If,  for  example,  in  a  line  \  mile  long,  the  surveyor  miss 
the  corner  70  links,  the  correction  is  made  as  follows: 

70  :  70  :  :  60  :  x  —  70  X  60  =  6CK  =  1°. 

3.  Measure  the  distance  between  the  extremity  of  this  random 
line  and  the  true  line.     Multiply  this  distance  by  57.3  and  divide 
the  product  by  the  length  of  the  line  surveyed.     The  quotient 
will  be  the  change  of  variation  expressed  in  degrees. 

Suppose  the  length  of  the  line  to  be  18.25  and  the  distance  be- 
tween the  extremities  35  links.  Then,  57.3X-35=20.055,  and 
20.055-^18  25=1.09°— 1°  §'  24".* 

4.  Multiply  the  distance  between  the  extremities  of   the  two 
lines  by  .3438  and  divide  the  product  by  the  length  of  the  line  to 
get  the  result  in  minutes. 

(  89 ).  In  determining  the  correction  to  be  made  in  the  follow- 
ing examples,  use  whichever  one  of  the  preceding  rules  appears 
most  expeditious : 

(1).  Length  of  line,  16  chains.  Distance  between  terminations 
18  links. 

(2).  Length  of  line,  36  chains.  Distance  between  extremities, 
32  links. 

(3).  Length  of  line,  20  chains.  Bearing  7°  32'.  Distance  be- 
tween extremities,  30  links. 

(4).  Length  of  line,  40  chains.  Distance  between  extremities, 
48  links.  Bearing  16°. 

(  90 ).  Let  us  now  see  whether  this  correction  to  be  made  is  to 
be  added  to  the  assumed  bearing  of  the  line,  or  subtracted  from  it. 

This  involves  two  cases : 

^Suppose  the  random  line  and  true  line  to  represent  two  radii  of  a  circle, 
and  the  line  connecting  their  extremities  to  be  a  portion  of  the  circumfer- 
ence of  the  same  circle.  We  shall  then  have  this  proportion: 

Whole  circumference  :  arc  :  :  360°  :  angle  of  lines. 

Whence,  in  case  under  consideration,  we  have 
(36.50  X  3.1416)  :  35  :  :  360°  :  1.09°. 

To  generalize,  let  A  represent  the  starting  point  of  the  lines  and  B  and  C 
their  extremities.  The  angle  formed  by  them  will  be  expressed  by  B  A  C. 

Then  2  A  B  X  3.1416  :  B  C  :  :  360°  :  B  A  C. 

Whence,  B  A  C  =  ?-|  X  57.3,  very  nearly 


56 


MANUAL   OF   PLANE  SURVEYING. 


1.  In  Northeast  and  Southwest  Courses.  RULE.— Add  the  minutes 
of  correction  to  the  assumed  bearing  of  the  line  (bearing  of  ran- 
dom line)  when  the  random  line  lies  to  the  left  of  the  true  line, 
and  subtract  when  it  lies  to  the  right.  The  sum  or  difference  will 
be  the  true  bearing  of  the  line. 

To  illustrate,  let  the  line  A  B,  Fig.  13,  represent  a  section  of  the 
true  meridian,  A  C  the  line  to  be  surveyed,  A  D  a  random  line 
lying  to  the  left,  and  A  E  a  random  line  lying  to  the  right  of  the 
true  line. 


Fio.  13. 

The  fact  is  apparent  that  the  bearing  of  the  line  A  C  is  equal  to 
the  bearing  of  the  line  A  D  plus  the  angle  D  A  C;  and  it  is 
equally  apparent  that  the  bearing  of  the  same  line,  A  C,  equals 
the  bearing  of  the  line  A  E  minus  the  angle  C  A  E. 

2.  In  Northwest  and  Southeast  Courses.  RULE. — Reverse  the  pre- 
ceding rale;  i.  e.,  subtract  the  minutes  of  correction  from  the  as- 
sumed bearing  (bearing  of  random  line)  when  the  random  line  lies 
to  the  left  of  the  true  line,  and  add  when  it  lies  to  the  right. 


MANUAL  OF  PLANE  SURVEYING.  57 

(91).  In  the  following  examples  the  bearings  of  the  random 
lines  are  given,  and  the  words  "  right "  and  "  left "  signify  their 
position  with  regard  to  the  true  line.  What  is  the  bearing  of  the 
true  line  in  each  case  ? 

(1).  N  40°  E  40.25,  left  .40. 
(2).  S  32°  W  16.50,  "  .25. 
(3).  N  71°  W  14.00,  right  .12. 
(4).  N  15°  30'  E  8.40,  "  .06. 
(5).  S  5°  15'  W  20.00,  "  .13. 
(6).  S  6°  17'  E  20.03, "left  .13. 
(7).  N  4°  06'  W  15.50,  "  .23. 
(8).  N  3°  21'  E  40.00,  right  .10. 

The  distances  are  all  given  in  chains  and  hundredths  of  a  chain. 

( 92  ).  So  far,  we  have  based  all  our  bearings  on  the  true  merid- 
ian, but  in  many  cases  lines  are  surveyed  and  their  magnetic  bear- 
ings, instead  of  their  true  bearings,  given. 

( 93 ).  When  this  is  the  case,  and  a  resurvey  is  to  be  made  of 
these  lines,  it  is  necessary  to  know  what  change  has  taken  place 
in  the  variation  of  the  needle  since  their  bearings  were  determined, 
because  the  bearings  change  with  the  variation.  Having  found 
this,  their  present  bearings  may  be  determined  by  the  following 
rule: 

1.  To  Correct  Magnetic   Bearings.— In  northeast  and   southwest 
courses,  add  the  change  to  the  given  bearing,  and  the  sum  will  be 
the   present  bearing.     In  northwest  and  southeast  courses,  sub- 
tract the  change  from  the  bearing,  and  the  difference  will  be  the 
present  bearing. 

2.  In  Fig.  14  let  A  B  and  A  C  represent  two  courses,  one  bear- 
ing northeast  and  southwest,  and  the  other  northwest  and  south- 
east.    Also,  let  A  D  represent  the  direction  of  the  needle  when  the 
lines  were  surveyed,  and  A  E  its  direction  at  the  present  time. 
The  change  of  variation  equals  the  angle  BAD,  and  it  will  be 
seen  that  the  present  bearing  of  the  line  A  C  is  equal  to  the  sum 
of  D  A  C  and  D  A  E.     Likewise,  that  the  present  bearing  of  the 
line  A  B  is  equal  to  the  difierence  between  DAB  and  DAE 

(  94  ).  This  rule  holds  good  while  the  needle  moves  toward  the 
west  If  its  movement  change,  the  rule  will  have  to  be  reversed. 


MANUAL  OF  PLANE  SURVEYING. 


FIG.  14. 

(  95 ).  Correct  the  bearing  of  each  of  the  following  courses  for 
the  present  year : 

(1).   N  14°  E,  annual  change  2'  30".     Survey  made  in  1856. 

(2).  N  7°  34'  W,  annual  change  3'.    Survey  made  in  1862. 

(3).   S  1°  13'  E,  annual  change  V.     Survey  made  in  1874. 

(4).   S  3°  02'  W,  annual  change  2'  15".     Survey  made  in  1858. 

( 96  ).  Sometimes  the  letters  T.  M.  are  placed  after  a  bearing  to 
denote  that  it  is  based  on  a  true  meridian,  and  M.  M.  sometimes 
follows  a  magnetic  bearing  to  show  that  it  is  based  on  the  mag- 
netic meridian.  These  precautions  are  necessary,  and  their  omis- 
sion is  frequently  a  cause  of  much  perplexity  to  surveyors. 

After  the  bearing  of  the  line  has  been  determined,  the  survey  of  it  is 
entirely  similar  to  the  survey  of  lines  whose  true  bearings  are  given, 

(  97  ).  When  the  surveyor  finishes  a  random  line,  he  generally 
walks  back  along  it  and  moves  all  the  stakes  to  the  true  line,  and 
perhaps  marks  trees  that  stand  on  or  near  the  true  line,  so  that  it 
may  be  the  more  easily  found  in  he  future. 


MANUAL  OF  PLANE  SURVEYING.  59 

He  then  enters  its  course  and  distance  in  his  field-notes,  or  sur- 
veyor's record,  and  this  completes  the  survey,  although  he  may 
sometimes  draw  a  plot  of  the  survey,  particularly  if  it  is  of  a  tract 
of  land. 

( 98  ).  Flag-men,  chain-men,  and  markers  are  usually  sworn 
before  the  survey  begins. 

(  99  ).  The  surveyor  should  be  careful  about  keeping  his  chain 
of  the  proper  length  by  testing  it  frequently  with  a  standard 
measure,  and  should  watch  his  assistants  closely  until  they  un- 
derstand what  is  required  of  them. 

It  would  also  be  well  for  every  county  to  have  a  true  meridian 
established,  so  that  the  variation  of  the  needle  might  be  found  at 
any  time,  and  with  but  little  trouble. 

( 100 ).  Back  sights  should  always  be  taken  at  intermediate 
stations  along  the  line  in  order  to  avoid  possibility  of  deflection. 

Having  set  off  the  change  of  variation  on  the  vernier,  the  lines 
of  the  tract  may  all  be  surveyed  without  moving  the  vernier  again. 
QUESTIONS  ON  CHAPTER  V. 

1.  What  is  a  "witness?" 

2.  How  are  witnesses  marked  ? 

3.  How  are  corners  found  from  witnesses  ? 

4.  What  is  meant  by  the  "course  and  distance "  of  a  line? 

5.  What  must  we  know  before  we  can  run  a  line? 

6.  How  do  you  set  off  east  variation  on  the  compass  vernier.' 

West? 

7.  Of  what  length  is  the  chain  usually  employed  in  surveying? 

8.  If  the  random  line  lies  to  the  right  of  the  true  line,  in  what 

direction  do  you  move  the  stakes? 

9.  Give  the  rule  for  correcting  the  stakes. 

10.  How  should  the  distance  between  the  extremities  of  the  true 

line  and  the  random  line  be  measured? 

11.  How  is  5  chains  and  12  links  usually  written  by  a  surveyor? 

16  chains  and  4  links? 

12.  Give  the  first  rule  for  finding  the  amount  of  error  in  as- 

sumed  bearing.     The  second. 

13.  What  is  meant  by  sine  f     Cosine  ? 

14.  Give  the  two  rules  for  correcting  assumed  bearings. 

15.  Give  the  rule  for  correcting  old  magnetic  bearings. 

16.  What  is  the  meaning  of  N  42°  E,  T  M  ?   S  14°  30'  W,  M  M? 

NOTK. — Observe  that  when  the  true  meridian  is  taken  as  the  basis  of  the 
survey,  the  variation  of  the  needle  is  set  off  on  the  vernier,  and  when  the  mag- 
netic "meridian  serves  as  the  basis,  the  change  of  variation  is  set  off  on  the 
vernier. 


CHAPTER  VI. 

UNITED  STATES  RECTANGULAR  SURVEYING. 

( 101 ).  The  statutes  of  the  United  States  provide  that,  "  The 
public  lands  shall  be  divided  by  north  and  south  lines  run  accord- 
ing to  the  true  meridian,  and  by  others  crossing  them  at  right 
angles,  so  as  to  form  townships  of  six  miles  square,  unless  where 
the  line  of  an  Indian  reservation,  or  of  tracts  of  land  heretofore 
surveyed  or  patented,  or  the  course  of  navigable  rivers  may  render 
this  impracticable ;  and  in  that  case  this  rule  must  be  departed 
from  no  further  than  such  particular  circumstances  require." 

( 102  ).  Again,  that,  "The  township  shall  be  divided  into  sec- 
tions, containing,  as  nearly  as  may  be,  six  hundred  and  forty  acres 
each,  by  running  through  the  same,  each  way,  parallel  lines  at  the 
end  of  every  two  miles,  and  by  marking  a  corner  on  each  of  such 
lines  at  the  end  of  every  mile." 

( 103  ).  There  are  many  other  important  provisions  relating  to 
the  subject  of  Public  Land  Surveying,  but  these  will  suffice.  We 
shall  now  see  how  they  are  carried  out. 

( 104 ).  The  fundamental  lines  upon  which  a  survey  is  based 
are  called  the  principal,  meridian  and  base  line.  The  first  of  these  is 
a  meridian  of  the  earth,  and  the  second  is  a  parallel  of  latitude. 
Their  point  of  intersection  is  called  the  initial  point.  Upon  these 
lines  every  piece  of  land  included  in  the  survey  has  a  direct  bear- 
ing, and  the  whole  survey  itself  is  located  by  the  number  or  name 
of  its  meridian.  For  instance,  the  position  of  a  small  tract  of  land 
in  Indiana  is  determined  by  its  distance  north  or  south  of  the  base 
line  and  east  or  west  of  the  principal  meridian,  but  the  survey  of 
nearly  the  whole  State,  as  well  as  of  other  contiguous  territory,  is 
governed  by  the  second  principal  meridian  which  runs  north  and 
south  a  short  distance  west  of  the  center  of  the  State.  In  like  man- 
(60) 


MANUAL  OF  PLANE  SURVEYING.  61 

ner,  the  survey  of  Ohio  is  based  upon  the  first  principal  meridian 
which  serves  as  the  western  boundary  of  the  State,  the  survey  of 
Michigan  is  regulated  by  the  Michigan  meridian,  and  the  surveys 
of  Minnesota  are  referred  to  the  fourth  and  fifth  principal  meri- 
dians. 

( 105  ).  The  selection  of  an  initial  point  is  the  first  step  in  the 
survey  of  any  new  territory,  and  this  is  always  chosen  at  some 
natural  and  imperishable  land-mark  found  in  or  near  the  lands  to 
be  surveyed.  From  this  point  the  principal  meridian  is  surveyed 
north  or  south,  or  north  and  south,  and  the  base  line  east  or  west, 
or  east  and  west.  Upon  these  lines,  which  are  surveyed  with  a 
fine  instrument  and  with  the  greatest  possible  precision,  six-miles 
distances  are  marked  for  township  corners,  one-mile  distances  for 
section  corners,  and  half-mile  distances  for  quarter  section  corners. 
Each  of  these  corners  is  marked  with  a  suitable  monument,  and 
appropriate  witnesses  are  also  chosen. 

(106).  From  each  six-mile  point  on  the  base-line,  east  and 
west  of  the  initial  point,  another  meridian  is  surveyed,  and  the 
territory  is  thus  divided  into  strips,  each  six  miles  wide,  lying 
north  and  south.  These  strips,  when  divided  into  townships,  are 
called  ranges.  The  first  one  east  of  the  principal  meridian  is  called 
range  1  east,  the  second  is  called  range  2  east,  the  third  range  3 
east,  and  so  on.  In  the  same  way  the  first  one  west  of  the  meridian 
is  called  range  1  west,  the  second  range  2  west,  etc. 

( 107 ).  Similarly,  lines  running  east  and  west  from  the  six- 
miles  points  on  the  principal  meridian,  divide  the  territory  into 
strips,  each  six  miles  wide,  lying  east  and  west.  The  meridians 
running  north  and  south  and  the  parallels  running  east  and  west 
thus  divide  the  territory  into  townships,  each  of  which  is  about  six 
miles  square,  and  consequently  contains  thirty -six  square  miles 
or  sections.  The  first  township  north  of  the  base-line  in  each 
range  is  called  township  1  north,  the  second  township  2  north,  and 
so  on ;  and  those  south  are  named  1  south,  2  south,  etc.,  to  the 
limit  of  the  survey.  The  first  township  north  of  the  base-line  and 
east  of  the  principal  meridian  is  described  as  township  1  north, 
range  1  east,  and,  in  a  similar  manner,  every  township  is  named 
with  regard  to  its  distance  from  the  base-line  and  from  the  princi- 
pal meridian.  This  is  conveniently  shown  in  Fig.  15. 

( 108  ).  Since  meridians  converge  as  they  approach  the  pole,  it 
a  evident  that  townships  can  not  be  quite  square,  and  that  every 


MANUAL   OF   PLANE  SURVEYING. 


township  must  be  somewhat  smaller  than  the  township  south  of  it 
and  larger  than  the  one  north  of  it,  except  in  certain  cases  on  the 
base-line.  In  the  northern  part  of  the  United  States  this  conver- 
gence is  greater  than  in  the  southern  part,  and  the  north  line  of  a. 


1 

s 

T,  3  N 
R.2E. 

Base 

T.I  N, 
R.2W. 

T.I  N. 
R,  1  E. 

Line. 

T,2  S. 
R.1W, 

T.2  S, 
R.2E. 

e 

J 

FIG.  15. 

township  in  some  places  is  more  than  one  hundred  feet  shorter 
than  the  south  line.  To  keep  the  error  arising  from  this  conver- 
gence within  reasonable  bounds,  lines  called  "correction  lines" 
have  been  surveyed  every  twenty-four  miles  or  four  townships  on 
the  north  side  of  the  base-line,  and  every  thirty  miles  or  five  town- 
ships south  of  the  base  line,  and  always  parallel  to  it.  Upon 
these  correction  parallels,  the  distances  are  measured  off  anew, 
same  as  on  the  base-line,  and  they  become  secondary  bases  in  the 


MANUAL   OF   PLANE   SURVEYING.  63 

survey,  although  townships  are  all  referred  to  the  base-line,  just 
as  if  they  did  not  exist. 

( 109 ).  For  convenience,  auxiliary  meridians  are  also  estab- 
lished every  eight  ranges  or  forty-eight  miles  east  and  west  of  the 
principal  meridian,  and  the  territory  is,  in  consequence,  divided 
into  rectangles  each  48  miles  long  by  24  miles  wide  north  of  the 
base-line,  and  48  miles  long  by  30  miles  wide  south  of  the  base-line. 

( 110).  The  manner  in  which  the  townships  are  surveyed  may 
easily  be  explained  by  Fig.  16.  Those  north  of  the  base-line  and 
east  of  the  principal  meridian  are  surveyed  by  commencing  at 
the  southeast  corner  of  township  1  north,  range  1  east,  and  run- 
ning north,  establishing  section  and  quarter-section  corners  at 
proper  distances,  four  hundred  and  eighty  chains,  or  six  miles,  to 
the  northeast  corner  of  the  township.  From  this  point  the  sur- 
veyor runs  west  six  miles,  or  four  hundred  and  eighty  chains,  to 
the  principal  meridian  and  finishes  the  survey  of  the  first  town- 
ship. He  then  surveys  the  next  township  north  in  exactly  the 
same  way,  and  continues,  as  indicated  by  the  numbers,  until  he 
closes  the  first  tier  of  townships  on  the  correction  parallel.  In  the 
same  way  he  begins  at  the  base-line  and  surveys  the  next  tier  east. 
The  townships  south  of  the  base-line  are  surveyed  in  the  same 
way,  except  that  the  surveyor  works  toward  the  base-line  instead  of 
from  it,  as  when  north,  as  shown  by  the  numbers. 

( 111 ).  West  of  the  principal  meridian  and  north  of  the  base- 
line the  survey  begins  at  the  southwest  corner  of  township  1  north, 
range  1  west,  and  proceeds  in  the  order  of  the  numbers.  South  of 
the  base-line  the  process  is  entirely  similar.  It  will  be  observed 
that  townships  east  of  the  principal  meridian  are  surveyed  by  run- 
ning first  north  and  then  west,  and  those  west  by  running  first  north 
and  then  eaxt. 

(  112  ).  Excesses  and  deficiencies  in  the  length  of  township  lines 
are  thrown  on  the  north  and  west  sides  of  the  townships  so  as  to 
fall  ultimately  into  the  north  and  west  tiers  of  sections.  These 
are  called  fractional  sections,  and  will  be  considered  in  due  time. 

( 113).  The  township  (congressional  township)  which  we  have 
had  under  consideration  must  not  be  confounded  with  the  civil 
township.  The  former  is  always,  when  not  fractional,  six  miles 
square,  but  the  latter  may  be  any  reasonable  size  or  shape  whatever. 

( 114 ).  Let  us  now  observe  how  the  townships  are  divided  into 
sections. 

NOTE  ON  ARTICLE  109.— In  later  surveys  the  rectangles  are  made  24  miles 
square  north  and  south  of  the  base-line. 


64 


MANUAL   OF   PLANE   SUKVEYING. 


CORRECTION    i           $ 

I             \PARALLEL 

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BASE 

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IS 

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1                |                I/'AA, 

FIG.  16. 


MANUAL  OF   PLANE  SURVEYING.  65 

1.  Before  the  surveyor  attempts  to  make  this  division,  he  ascer- 
tains the  bearings  of  the  boundary  lines  of  the  township,  so  as  to 
be  able  to  run  the  interior  lines  as  nearly  parallel  to  them  as  pos- 
sible. 

2.  These  bearings  may  generally  be  easily  obtained  from  the 
notes  of  the  previous  survey,  according  to  the, directions  we  have 
already  given,  but  he  may  also  obtain  them  by  retracing  a  section 
of  one  or  more  of  the  exterior  lines  of  the  township  and  determin- 
ing the  bearing  from  the  line  itself.     He  should  also  measure  sec- 
tions of  the  township  line  to  see  how  his  chain  agrees  in  length 
with  the  chain  used  in  the  previous  survey.     Having  attended  to 
these  preliminaries,  he  is  ready  to  begin  work. 

3.  The  townships,  as  we  have  before  stated,  contain  thirty-six 
sections  each,  and  these  sections  are  numbered,  as  shown  in  Fig.  17. 


90 

68 

5'l                   |  34 

"7 

6 

5 

4 

3 

2 

1 

39 

67 

50 

31 

16 

86            87 

65            66 

4*           49 

3"             i* 

.4            -.5 

7 

8 

9 

to 

11 

12 

34 

85 

<H 

47 

30 

13 

82            33 

62        63 

45            46 

28             39 

1                       12 

18 

17 

16 

15 

14 

13 

BO 

81 

61 

44 

27 

10 

78            -79 

69            60 

42             43 

25                 2J6 

8                9 

19 

20 

21 

22 

23 

24 

76 

77 

58 

41 

24 

7 

74              75 

56             57 

39            40 

22                   23 

5                C 

30 

29 

28 

27 

26 

25 

72  • 

73 

55 

38 

2t> 

4 

70              71 

53             54 

36            s 

15                   20 

2                 3 

31 

32 

33 

34 

35 

36 

63 

5? 

35 

iS 

« 

FIG.  17. 


66  MANUAL   OF   PLANE  SURVEYING. 

The  manner  in  which  the  sections  are  surveyed  will  now  be  ex- 
plained. 

4.  The  surveyor  goes  to  the  south-west  corner  of  section  thirty- 
six  to  begin  the  survey.     At  this  point  he  sets  his  compass  at  the 
bearing  of  the  east  line  of  the  township,  which  should  be,  but  sel- 
dom is,  the  true  meridian,  and  runs  north  forty  chains.     Here  he 
establishes  a  quarter-section  corner  between  sections   thirty-five 
and  thirty -six,  and  then  proceeds  forty  chains  or  one- half  mile 
further  to  the  corner  of  the  section,  or  rather  to  the  corner  of  sec- 
tions twenty-five,  twenty-six,  thirty-five  and  thirty-six,  since  these 
four  sections  have  a  common  corner  here.     Distances  from  the 
starting  point  at  which  brooks,  creeks  and  other  objects  of  im- 
portance are  met  on  the  line,  are  carefully  noted.     From  this  cor- 
ner he  runs  a  random  line  to  the  east  line  of  the  township.     If 
this  line  intersects  the  township  line  at  the  first  mile  corner,  it  is 
marked  back  as  the  true  north  line  of  section  thirty-six,  but  if  it 
does  not,  the  distance  which  it  misses  the  corner,  either  right  or 
left,  is  noted,  and  the  line  changed  accordingly. 

5.  Having  returned  to  the  north-west  corner  of  section  thirty- 
six,  he  next  proceeds  to  survey  section  twenty-five  in  the  same 
manner,  and  he  follows  the  route  indicated  by  the  figures  until  he 
completes  the  survey  of  the  eastern  tier  of  sections.     It  will  be  ob- 
served that  when  he  completes  the  survey  of  section  twelve,  he 
then  finishes  up  the  survey  of  section  one  by  running  north  on  a 
random  line  and  correcting  back  to  the  south-west  quarter. 

6.  He  next  surveys  the  second  tier  of  sections  by  beginning  at 
the  south-west  corner  of  section  thirty-five,  and  proceeding  north 
in  the  same  manner  as  in  the  survey  of  the  first  tier,  and  thus  he 
continues  until  he  reaches  the  fifth  tier.     Here,  after  surveying 
section  thirty-two,  he  runs  west  from  its  north-west  corner  to  the 
range  line  or  meridian  and  completes  the  survey  of  section  thirty- 
one,  and  continues  to  work  north,  surveying  the  fifth  and  sixth 
tiers  together,  until  he  reaches  the  north  line  of  the  township. 

(115).  The  township  is  now  divided  into  sections,  each  of 
which,  except  those  in  the  north  and  west  tiers,  called  fractional 
sections,  is  sold  as  containing  six  hundred  and  forty  acres  of  land, 
more  or  less,  and  each  quarter-section  as  containing  one  hundred 
and  sixty  acres,  more  or  less.  The  fractional  sections  generally 
contain  a  greater  or  less  quantity  of  land  than  the  other  sections, 
because  all  excesses  and  deficiencies  fall  to  them.  In  surveying 


MANUAL   OF   PLANE   SURVEYING.  67 

them, -however,  the  quarter-section  corners  between  them  are  so 
placed  that  the  excesses  and  deficiencies  fall  to  the  exterior  quar- 
ters, and  the  interior  quarters,  or  those  touching  the  other  sections 
of  the  township,  are  sold  as  containing  the  proper  amount  of  land — 
one  hundred  and  sixty  acres  each.  The  exterior  quarters  are  sold 
as  containing  whatever  the  measurements  of  the  survey  indicate 
that  they  contain.  Section  six  is  sometimes  called  the  "double 
fractional,"  and  usually  contains  only  one  exact  quarter. 

(116).  Whenever,  in  the  course  of  a  survey,  an  impassable  bar- 
rier, such  as  a  lake  or  navigable  river,  is  met,  the  surveyor  estab- 
lishes what  is  called  a  meander  corner  on  its  margin,  and  then 
runs  a  meander  line  from  this  corner  along  the  edge  of  the  obsta- 
cle. The  rivers  and  lakes  thus  meandered  are  reserved  in  the  sale 
of  the  public  lands. 

(117).  The  surveyor,  from  the  time  the  survey  of  the  principal 
meridian  is  begun  until  the  township  is  divided  into  sections, 
marks  every  half-mile  of  true  line  that  he  surveys  with  a  corner, 
and  keeps  an  account  in  his  field-notes  of  every  important  object 
he  meets  in  the  survey,  as  well  as  a  topographical  description  of 
the  country.  He  also  makes  two  sets  of  corners  on  the  correction 
parallels,  one  for  townships  north,  and  the  other  for  townships 
south  of  the  line.  Aside  from  this  it  is  not  unusual  to  find  two 
sets  of  corners  on  interior  parallels  and  meridians,  owing  to  dis- 
crepancies between  contiguous  surveys. 

(118).  The  monuments  used  in  marking  corners  are  always 
adapted  to  the  country  in  which  the  survey  is  made,  and  their 
position  is  generally  witnessed  by  one  or  more  bearing  trees,  or 
mounds  of  earth  thrown  up  around  a  stake  or  a  stone,  whose 
courses  and  distances  from  the  corner  are  carefully  noted  in  the 
field-books  of  the  survey.  These  witnesses,  if  trees,  are  always 
marked  facing  the  corner,  which  enables  them  to  be  more  easily 
found  at  any  subsequent  time. 

(119).  This  completes  the  work  of  the  Government  Surveyor, 
or  deputy,  as  he  is  usually  called.  He  returns  his  notes  to  the 
Surveyor  General  of  his  district,  and  these  notes  become  the  basis 
of  all  subsequent  surveys.  The  work  of  dividing  and  sub-dividing 
the  sections,  which  belongs  to  the  county  and  private  surveyors, 
we  shall  consider  in  the  next  chapter. 

(120).  This  beautiful  system  of  land  surveying,  not  unlike  the 
old  Roman  system,  was  devised  about  the  year  1785,  for  the  pur- 

NOTE  ON  ARTICLE  117.— In  a  few  instances  three  sets  of  corners  have  been 
established  on  range  lines,  but  at  the  present  time  only  one  set  is  established 
except  on  the  base-line  and  correction  parallels. 


MANUAL   OF  PLANE  SURVEYING. 

pose  of  preparing  the  North  West  Territory  for  settlement,  and 
has  answered  the  purpose  in  an  admirable  manner.  It  is  the  re- 
sult of  mature  deliberation,  and  exhibits  no  mean  knowledge  of 
engineering  skill,  and,  like  many  other  great  inventions,  its  beauty 
and  utility  consist  in  its  extreme  simplicity.  It  has  long  since 
outgrown  the  limits  for  which  it  was  intended,  and  soon  nearly 
the  whole  territory  between  the  western  boundary  of  Pennsylvania 
and  the  Pacific  ocean  will  be  united  in  one  complete  net-work  of 
sections. 

The  readiness  with  which  it  enables  a  surveyor  to  re-trace  old 
lines  and  determine  the  location  of  lost  corners  prevents  an  end- 
less amount  of  litigation  common  to  States  not  surveyed  accord- 
ing to  this  system. 

QUESTIONS  ON  CHAPTER  VI. 

1.  What  are  the  fundamental  lines  of  a  survey? 

2.  What  is  their  point  of  intersection  called? 

3.  Upon  what  principal  meridian  is  the  survey  of   Indiana 

based? 

4.  What  is  a  range?     A  township? 

5.  Draw  a  diagram  representing  town  4  south,  range  3  east. 

6.  How  is  the  error  caused  by  the  convergence  of  the  merid- 

ians arrested? 

7.  How  are  townships  north  of  the  base-line  and  east  of  the 

principal  meridian  surveyed  ?     South  of  the  base-line  and 
west  of  the  principal  meridian? 

8.  What  is  the  difference  between  a  congressional  township 

and  a  civil  township? 

9.  How  may  a  surveyor  ascertain  the  bearing  of  the  lines  that 

bound  a  township? 

10.  How  many  sections  in  a  township?     How  are  they  num- 

bered? 

11.  Where  does  a  surveyor  begin  work  when  he  divides  a  town- 

ship into  sections  ? 

12.  Describe  the  method  of  surveying  the  eastern  tier  of  sec- 

tions.    The  western. 

13.  Name  the  fractional  sections.   The  double-fractional  section. 

14.  What  causes  fractional  sections? 

15.  How  many  acres  in  a  section? 


MANUAL   OF   PLANE   SURVEYING.  69 

16.  What  quarters  of  fractional  sections  are  generally  full? 

17.  What  is  a  meander  line? 

18.  Why  are  two  sets  of  corners  needed  on  correction  lines? 

19.  How  far  apart  are  the  corners  on  lines  surveyed  by  the  gov- 

ernment surveyor? 

20.  For  what  purpose  was  the  rectangular  system  devised? 

When? 

NOTE  ON  CHAPTER  VI. — The  system  of  surveying  described  in  this  Chapter 
was  not  set  forth  in  its  present  completeness  at  the  beginning.  The  first 
act  of  Congress  on  the  subject  was  that  of  May  20,  1786,  and  under  this  act 
the  first  public  surveys  were  made.  The  territory  covered  by  these  surveys 
forms  a  part  of  the  State  of  Ohio,  and  is  generally  called  the  "  Seven 
Ranges  "  In  it  the  sections  are  numbered  north  and  south  from  the  south- 
east corner  of  the  township.  The  Geographer  of  the  United  States  directed 
the  surveys.  Although  the  principles  of  the  system  have  virtually  re- 
mained unchanged,  the  system  itself  has  been  made  definite  and  concise 
since  iis  inception. 


CHAPTER    VII. 

THE  DIVISION  AND  SUB-DIVISION  OF  THE  SECTION. 

( 121).  The  county  or  other  surveyor  makes  all  his  surveys  in 
accordance  with  the  field-notes  of  the  original  survey,  a  copy  of 
which  forms  a  part  of  the  public  records  of  each  county. 

Subsequent  surveys  may  prove  that  great  irregularities  exist  in 
the  original  survey,  but  none  of  the  lines  or  corners  can  be  changed. 

(122).   The  ideal  section  of  the  young  surveyor  frequently 


3  If 


FIG.  18. 

differs  greatly  from  the  real  section  he  meets  in  practice.  The 
ideal  section  is  very  nearly  a  perfect  square — varying  a  little  on 
account  of  the  convergence  of  the  meridians ;  it  is  bounded  by 
four  straight  lines,  and  contains  almost  exactly  640  acres  of  land. 

(70) 


MANUAL   OF   PLANE   SURVEYING.  71 

The  real  section  may  sometimes  be  far  from  square;  its  boundary 
lines  may  deflect  every  half-mile  on  its  perimeter,  and  its  area 
may  exceed  by  several  acres  the  area  of  another  section  adjoining. 
The  surveyor,  however,  must  adhere  as  closely  as  possible  to  the 
original  survey,  and  let  the  section  and  divisions  and  sub-divisions 
of  the  section  contain  whatever  the  government  deputy  saw  proper 
to  put  into  them. 

(123).  Let  us  now  see  what  the  principal  divisions  and  sub- 
divisions of  the  section  are,  and  then  consider  the  method  of  sur- 
veying them,  locating  the  corners,  etc. 

(124).  Fig.  18  represents  a  section,  with  some  of  the  main  di- 
visions and  sub  divisions  laid  off  on  its  face.  They  are  described 
.as  follows : 

1.  S.  W.  qr. 

2.  N.  W.  qr. 

3.  S.  }  N.  E.  qr. 

4.  E.  \  S.  E.  qr. 

5.  W.  }  S.  E.  qr. 

6.  N.  E.JN.  Kqr. 

7.  S.  J  N.  W.  J  N.  E.  qr. 

8.  N.  E.  i  N.  W.  i  N.  E.  qi. 

9.  N.  W.  \  N.  W.  \  N.  E.  qr. 

Of  these  tracts,  Nos.  1  and  2  each  contain  160  acres;  3,  4  and  5 
each  80  acres ;  6,  40  acres ;  7,  20  acres ;  and  8  and  9  each  10  acre? 

( 125  ).  Whenever  an  instrument  of  writing,  such  as  a  deed  or 
mortgage,  implying  responsibility  to  the  amount  of  the  descrip- 
tion, bears  on  a  tract  of  laud,  the  area  is  usually  qualified  by  the 
compound  term  "  more  or  less."  For  instance,  No.  1,  above,  would 
be  described  as  containing  160  acres,  more  or  less ;  No.  3,  as  con- 
taining 80  acres,  more  or  less,  and  so  on. 

( 126 ).  The  corners  of  the  section,  or  of  the  different  parts  of  it, 
are  named  from  their  position.  The  principal  ones  are  the  sec- 
lion  corners,  J  corners,  £  \  corners,  and  J-  J  corners. 


72 


MANUAL  OF  PLANE  SURVEYING. 


The  numbers  on   the  diagram  correspond  with  the 
names  after  similar  numbers  below. 


to 


!l 


£ 

a          / 

S            Z 

3 

/              3 

1 

3 

z 

S          2 

0          B 

IS  7  If 

FIG.  19. 

1.  Section  Corners : 

(1).  N.  E.  Corner. 
(2).  S.  E.      " 
(3).  S.  W.     " 
(4).  N.W.     " 

2.  Quarter-Section  Corners: 

(5).  N.  \  Corner. 
(6).  E.  \        « 
(7).  S.  \        " 


(9).  Center 


MANUAL  OF  PLANE  SVRVEYING.  73 

3.  Half-Quarter  Corners: 

(10).  N.  H  W.  Corner. 

(11).  N.  HE.  « 

(12).  E.  i  i  K  " 

(13).  E.  £  \  S.  " 

(14).  S.   J  i  E.  « 

(15).  S.   J  i  W.  « 

(16).  W.H&  '• 

(17).  W.H>'.  " 

(18).  N.  H 

(19).  E.  H 

(20).  S.    H 

(21).  W.H 

4.  Fourth-Quarter  Corners: 

(22).  N.  W.  Center  Corner. 
(23).  N.  E.        "          " 
(24).  S.  E. 
(25).  S.  W.       " 

( 127  ).  The  exterior  lines  of  the  section  are  called  section  lines, 
and  the  two  lines  that  cross  at  the  center  of  the  section  are  called 
center  lines. 

( 128 ).  We  are  now  ready  to  examine  the  method  by  which  the 
position  of  each  of  the  various  classes  of  corners  is  determined. 

1.  The  section  corners,  and  the  exterior  quarter-section  corners, 
except  in  an  occasional  case  on  a  town  or  range  line,  are  set  by  the 
Government  deputy,  and  the  surveyor  who  follows  him  sets  the  re- 
maining quarter-section  corner  (center)  and  all  the  minor  corners. 

2.  The  section  is  divided  and  corners  set  according  to  instruc- 
tions from  the  proper  authority,  and  the  following  is  the  author- 
ized method  of  setting  the  center  corner  : 

(1).    A  line  is  run  connecting  the  N.  J  corner  with  the  S  }  cor- 


74  MANUAL   OF   PLANE   SURVEYING. 

ner,  and  another  connecting  the  E.  |  corner  with  the  W.  £  corner. 
The  point  of  intersection  of  these  two  lines  is  taken  for  the  center 
of  the  section. 

The  preceding  is  the  method  in  general  use,  and  is  perhaps  the 
most  equitable  one  that  could  be  devised,  but  the  following  has 
been  used  in  some  places  : 

(2).  A  line  is  surveyed  connecting  the  E.  £  corner  with  the  \V.  } 
corner,  and  the  middle  point  of  this  line  is  taken  for  the  center 
of  the  section. 

If  the  section  lines  were  straight  from  one  corner  of  the  section 
to  the  other,  and  the  quarter-section  corner  midway  between  the 
section  corners,  the  corner  determined  by  this  method  would  coin- 
cide in  position  with  the  one  determined  by  the  other;  but  as  this 
is  not  always  the  case,  they  may  differ  considerably  in  position. 

3.  To  Set  a  Half-Quarter  Corner. — Kun  a  line  along  the  quarter- 
section,  on  the  side  upon  which  the  corner  is  to  be  set,  and  from 
one  corner  to  the  other.     Bisect  this  line  for  the  corner. 

For  example :  To  set  the  E.  J  $  corner,  we  connect  the  center 
corner  and  E.  J  corner  with  a  line,  and  the  middle  point  of  this 
line  is  the  required  corner. 

4.  To  Set  a  Fourth- Quarter  Corner.  -First  set  a  i  }  corner  on  each 
of  two  opposite  sides  of  the  quarter-section.     Then  connect  these 
two  corners  with  a  line,  and  bisect  this  line  for  the  }  £  corner. 

To  illustrate :  In  order  to  set  the  N.  E.  center  corner,  it  is  nec- 
essary first  to  set  the  N.  £  i  E.  corner  and  the  E.  i  J-  corner,  or  the 
E.  J  J  N.  corner  and  the  N.  £  }  corner.  The  middle  point  of  a 
line  connecting  one  corner  with  the  other,  in  either  set,  for  it  is 
immaterial  which  is  taken,  will  be  the  required  corner. 

5.  The  same  methods  are  employed  for  setting  corners  to  the 
divisions  of  the  fourth  of  quarter-sections.     For  instance,  to  set 
the  north-east  corner  of  the  S.  $  N.  W.  J  N.  E.  qr.,  it  is  necessary 
only  to  bisect  the  line  between  the  N.  E.  center  corner  and  N.  £  £ 
E.  corner. 

EXAMPI/2S. 

(  129 ).    (1).   How  would  you  set  the  S.  H  corner? 
(3).   How  is  the  S.  J  J  W.  corner  set? 
(4).   Describe  the  method  of  setting  the  S.  W.  center 
corner. 

( 130).  We  are  now  prepared  to  understand  how  the  divisions 
and  sub-divisions  of  the  section  are  surveyed. 

1.   A  quarter-section  is  surveyed  by  running  the  two  exterior 


MANUAL   OF   PLANE   SURVEYING.  75 

Lalf-mile  lines,  and  the  two  center  lines  of  the  section,  in  order  to 
locate  its  interior  corner  and  fix  its  interior  boundaries. 

2.  A  half-quarter  is  surveyed  by  first  surveying  as  many  of  the 
lines  of  the  quarter  as  enter  wholly  or  in  part  into  its  boundaries, 
then  whatever  other  lines  are  necessary  to  determine  its  remain- 
ing corners,  and  finally  the  lines  connecting  these  corners  with 
one  another,  or  with  others. 

For  instance,  to  survey  the  S.  £  S.  E.  qr.,  it  would  be  necessary 
to  run  the  south  line  of  the  quarter,  the  east  line,  the  two  center 
lines  of  the  section,  and  the  E.  and  W.  center  line  of  the  south- 
east quarter. 

3.  A  fourth-quarter  is  surveyed  on  the  same  principle  as  the 
half-quarter. 

For  example,  in  surveying  the  S.  E.  J  S.  E.  qr.,  it  is  necessary 
to  survey  the  south  and  east  lines  of  the  quarter,  the  E.  and  W. 
and  N.  and  S.  center  lines  of  the  section,  either  the  E.  and  W.  or 
the  N.  and  S.  center  line  of  the  S.  E.  qr.,  and  finally  the  north  line 
of  the  fourth-quarter,  if  the  N.  and  S.  center  line  has  been  sur- 
veyed, or  the  west  line,  if  the  E.  and  W.  center  line  has  been  sur- 
veyed. 

( 131 ).  As  a  general  rule  for  the  survey  of  a  tract  of  land  bear- 
ing a  relation  to  the  section,  the  following  is  submitted :  Run 
lines  to  connect  the  known  corners  of  the  tract  when  no  unknown 
corner  intervenes  between  them,  then  such  lines  as  are  necessary 
to  determine  the  location  of  unknown  corners,  and  finally  lines  to 
connect  these  newly  located  corners  with  one  another,  or  with 
others. 

( 132  ).  In  many  cases  some,  if  not  quite  all,  the  boundary  lines 
of  a  tract  of  land  are  known.  There  is  seldom  any  need  of  re- 
surveying  these,  except  where  it  must  be  done  in  order  to  establish 
other  lines  or  corners. 

EXAMPLES. 

(133).  What  lines  would  a  surveyor  have  to  run  in  order  to 
establish  all  the  boundaries  of  each  of  the  following  described 
tracts? 

(1).    S.  W.  qr. 

(2).   N..JN.  E.  qr. 

(3).    S.  }  N.  W.  qr. 

(4).   N.  E.  J  S.  W.  qr. 

(5).   N.W.JN.W.qr. 


/6  MANUAL    OF    PLANE   SURVEYING. 

( 134  ).  So  far  we  have  dealt  exclusively  with  corne-s,  lines, 
and  tracts  which  may  be  called  independent : 

The  corners,  because  their  position  is  determined  by  the  division 
of  certain  lines,  and  is  not  definitely  fixed,  so  far  as  distance  is 
concerned,  by  any  other  point  in  the  section. 

The  lines,  because  they  connect  the  corners,  and  may,  therefore, 
vary  in  a  limited  degree  either  in  course  or  distance,  or  both. 

The  tracts,  because  they  are  limited  by  the  lines  and  are  not 
definitely  fixed  as  to  area  or  figure. 

( 135 ).  To  illustrate  the  preceding  still  further,  suppose  the 
north  line  of  a  quarter-section  to  be  36  chains  long,  instead  of  40 
chains  long,  and  the  contents  of  the  quarter-section  to  be  148 
acres,  instead  of  160  acres.  The  J  J  corner  on  the  north  line  will 
then  be  18  chains  from  the  section  corner,  and  the  same  distance 
from  the  \  corner,  and  each  of  the  lines  will  be  but  18  chains  in 
length,  and  a  fourth-quarter  out  of  the  quarter-section  may  fall 
short  three  or  four  acres.  If,  now,  the  north  line  of  the  quarter- 
section  had  been  longer,  the  N.  5  \  corner  would  have  been 
further  from  each  of  its  north  corners,  and  the  area  of  the 
fourth-quarter  would  have  been  greater. 

(136).  There  is  a  class  of  corners,  lines,  and  tracts,  however, 
that  may  be  called  dependent : 

The  corners,  because  their  distance  is  fixed  from  some  given 
point,  and  is  not  obtained  by  bisecting  a  line. 

The  lines,  because  they  connect  the  corners,  and  are  conse- 
quently of  a  definite  course  and  distance. 

The  tracts,  because  they  are  bounded  by  the  lines,  and  their 
area,  therefore,  is  not  affected  by  the  excess  or  deficiency  of  land 
in  the  section. 

Dependent  corners,  then,  are  those  whose  position  is  definitely 
fixed ;  dependent  lines  connect  dependent  corners,  and  dependent 
tracts  are  bounded  by  dependent  lines. 

(137).  Dependent  lines  and  tracts  are  surveyed  without  any 
reference  whatever  to  the  division  or  sub-division  of  the  section, 
although  they  may  depend  on  some  corner  in  the  section  as  a  base. 

A  tract  may  be  partly  dependent  and  partly  independent. 

( 138).  The  following  are  examples  of  descriptions  of  depend- 
ent tracts: 

(1).  N.  45°  E.  16.00;  thence  N.  45°  W.  10.00;  thence  S.  45°  W. 
16.00;  thence  S._45°  E.  10.00,  to  the  place  of  beginning. 


MANUAL   OF   PLANE   SURVEYING.  77 

(2).  Commencing  at  the  S.  E.  corner  of  section  22,  town  3  N., 
range  4  E.  and  running  thence  N.  15°  E.  12.00;  thence  S.  45°  E. 
12.00;  thence  S.  75°  W.  12.00,  to  the  place  of  beginning. 

( 139).  In  a  dependent  tract  the  course  and  distance  of  each  of 
its  boundary  lines  are  usually  given  in  the  description  of  it,  and 
it  is  then  said  to  be  described  by  "  metes  and  bounds." 

( 140).  The  rules  given  for  setting  corners,  running  lines,  and 
surveying  tracts,  in  full  sections,  also  apply  to  fractional  sections, 
except  where  their  fractional  sides  do  not  contain  half,  or  much 
more  than  half,  the  usual  amount  of  land  found  in  these  parts 
of  sections,  or  where  the  amount  they  contain  considerably  ex- 
ceeds the  usual  amount.  In  the  first  case,  the  outside  tier  of 
fourths  of-quarter  is  omitted,  and  in  the  second  the  excess  is 
usually  thrown  to  them,  and  the  inside  tier  of  fourths-of -quarter 
in  the  outside  quarters  of  the  sections,  are  left  of  their  usual  size. 
If  the  deficiency  is  very  great,  perhaps  the  entire  outside  tier  of 
quarters  is  wanting.  By  act  of  Congress  approved  April  24, 1820, 
fractional  sections  containing  less  than  160  acres  are  not  to  be  di- 
vided in  the  original  survey.  In  the  division  of  fractional  sec- 
tions, generally,  such  rules  must  be  adopted  as  the  exigencies  of 
the  case  seem  to  require. 

( 141 ).  When  a  tract  of  land  lies  partly  in  one  quarter  of  a 
section  and  partly  in  another,  each  part  should  generally  be  de- 
scribed and  surveyed  separately ;  and  the  same  may  be  said  of 
tracts  extending  into  two  or  more  sections,  and  sometimes  also  of 
those  extending  into  different  fourths  of  the  same  quarter.  The 
following  are  examples : 

(1).   N.  }  N.  E.  qr.,  and  N.  E.  J  N.  W.  qr. 

(2).   S.  W.  i  N.  E.  qr.,  and  N.  J  N.  W.  J  S.  E.  qr 

(3).   N.  E.  qr.  Sec.  4,  and  N.  W.  qr.  Sec.  3. 

(4).   S.  E.  i  S.  E.  qr.  Sec.  35,  and  W.  \  S.  W.  qr.  Sec.  36. 

(5).  S.  E.  J  N.  W.  qr.,  and  E.  }  N.  E.  \  N.  W.  qr. 

In  some  of  the  cases  an  occasional  line  may  be  a  boundary  to 
each  part  of  the  tract,  as  the  east  line  of  No.  5,  described  above. 

QUESTIONS  ON  CHAPTER  VII. 

1.  Does  the  section  always  contain  exactly  640  acres? 

2.  Draw  a  diagram  of  a  section  and  represent  the  following 

tracts  on  it :     N.  E.  qr.;  N.  W.  }  N.  W.  qr.;  S  \  S.  W.  qr.; 
N.  }  N.  E.  }  N.  W.  qr. 

3.  Why  is  the  phrase  "  more  or  less "  used  in  descriptions  of 

land  in  deeds,  etc.? 


78  MANUAL   OF   PLANE   SURVEYING. 

4.  Name  the  quarter-section  corners.     The  J  \  corners.     The 

J  }  corners. 

5.  What  are  the  "  center  lines  "? 

6.  What  eight  corners  does  the  Government  deputy  establish 

to  nearly  every  section  ? 

7.  Does  he  ever  set  an  interior  corner  of  a  section? 

8.  Give  the  first  method  of  finding  the  center  of  a  section? 

9.  Prove  that  the  first  and  second  methods  would  agree,  if  the 

section  lines  were  straight  from  one   section   corner  to 
another. 

10.  How  do  you  set  a  \  \  corner?     A  \  \  corner? 

11.  How  would  you  survey  a  quarter  section?     A  half-quarter? 

A  fourth-quarter? 

12.  Give  a  general  rule  for  the  survey  of  a  tract  of  land  bearing 

a  relation  to  the  section. 

13.  What  is  the  difference  between  independent  and  dependent 

corners?     Lines?     Tracts? 

14.  Describe  an  independent  tract.     A  dependent  tract. 

15.  When  do  the  general  methods  for  dividing  and  sub-dividing- 

sections  not  hold  good  in  fractional  sections? 

16.  When  should  different  parts  of  a  tract  be  surveyed  sepa- 

rately? 


CHAPTER  VIII. 

FIELD-NOTES. 

( 142 ).  The  field-notes  of  sectional  surveys  by  the  Government 
deputy  show : 

1 .  The  witnesses  taken  at  section  and  quarter-section  corners. 

2.  The  length  of  the  fractional  lines  in  fractional  sections. 

3.  The  number  of  acres  in  each  of  the  fractional  quarters  of 
fractional  sections. 

4.  The  offsets  between  section  corners  in  one  township  and  the 
corresponding  ones  in  the  adjoining  township.     These  offsets  are 
sometimes  found  on  town  and  range  lines  as  well  as  on  correction 
parallels. 

5.  The  distances  from  the  starting  point  of  a  line  at  which 
brooks  and  creeks  are  crossed,  and  trees  and  other  objects  met  with 
on  the  line. 

6.  A  description  of  the  timber,  surface,  soil,  etc. 

7.  The  courses  and  distances  of  meander  lines  surveyed  along 
rivers,  lakes,  etc. 

(143).  Each  full  section  is  supposed  to  contain  640  acres  of 
land,  and  it  is  always  taken  for  granted  that  the  distance  between 
a  section  corner  and  a  quarter-section  corner  is  40  chains.  The 
lines  are  also  supposed  to  run  due  north  and  south  and  east  and 
west.  These  suppositions,  however,  are  strictly  correct  only  in 
comparatively  few  cases. 

Quarter-section  corners,  like  section  corners,  on  town  and  range 
lines  answer  for  sections  on  each  side  of  the  line,  but  where  an 
offset  occurs  aud  the  closing  section  corner  is  set  either  at  one  side 
or  the  other  of  the  corner  already  on  the  line,  the  quarter-section 
corner  for  the  closing  section  is  omitted.  In  this  case  the  closing 
section  has  only  seven  corners  instead  of  eight. 

NOTE  ON  ARTICLE  (142).—  The  field-notes  of  several  .-tates  have  been  turned 
over  to  the  State  authorities,  and  may  be  procured  from  them  when  needed. 

(79) 


80 


MANUAL   OF   PLANE   SURVEYING. 


( 144  ).  For  convenience  of  reference  a  plot  of  the  township  is 
drawn  after  the  survey  is  completed,  and  whatever  is  essential  to 
subsequent  surveys  is  represented  on  it. 

(  145 ).  Fig.  20  will  give  an  idea  of  the  manner  in  which  a  plot 
of  this  kind  is  drawn,  although  space  will  not  permit  its  being 
made  as  complete  as  it  should  be. 


£ 

A 
J 

I 

^ 

/ 
K 

L 

l 


6    < 

3        5       < 

>    4    * 

>    3    c 

>    2     < 

7    > 

>    8    > 

n       9         i 

r»     10     i 

i    11     i 

>    12 

18    ' 

••   17     • 

*•  (6    - 

*•  15    • 

i-    14-    • 

*    13 

19    , 

j    20    < 

o  21     < 

i  22    t 

,    23   * 

•>    24 

30    c 

a    29     , 

j    28       r 

j  27    , 

•   2G    < 

j    25 

3!     - 

-    32    - 

-  33    - 

-   34  - 

-    35    - 

-    36 

w; 


u 

T 


FIG.  20. 

The  capital  letters  on  the  margin  designate  the  section  corners 
on  the  town  and  range  lines,  and  the  small  letters  the  quarter- 
section  corners. 

The  interior  section  corners  are  designated  by  the  numbers  of 
sections.  Thus,  the  southwest  corner  of  section  one  is  numbered 


MANUAL  OF  PLANE  SURVEYING.  81 

1,  2,  11,  12,  because  it  serves  as  a  corner  to  each  of  these  sections. 

The  interior  quarter-section  corners  are  designated  as  corners  1 
to  6,  respectively,  on  the  line  B  E,  I  W,  or  whatever  line  it  may  be. 

( 146  ).  Attached  to  the  plot  of  the  township  is  a  list  of  the  wit- 
nesses at  each  of  the  corners,  generally  on  the  principle  of  the  fol- 
lowing : 

1.  Exterior  corners. 

Sections. 

At  A.   Be  6  N  14  W  32,  Ash  16  S  12  W  17. 
At  B.   Oak  10  S  19  E  11,  Hickory  18  N  5  W  6. 
At  C.   Maple  14  N  12  E  41,  Poplar  28  S  72  W  19. 
Qu  arter-sections. 

At  a.   Oak  14  N  61  W  14,  Sugar  15  S  16  W  13. 
At  b.   Elm  26  S  12  E  25,  Ash  9  S  63  W  14. 

2.  Interior  corners. 

Sections. 

(1).  Cor.ofl,2,ll,12,Maplel5N10E19,Elml6S71E12. 

(2).  Cor.  of  2, 3, 10, 11,  Oak  36  S 15  W 12,  Ash  20  N  82  W41. 
On  the  line  B  R  (Quarter-section  corners). 

(1).  At  1.  Be.  12  N  16  E  42,  Gum  14  S  15  W  22. 

(2).  At  2.  Pop.  28  S.  46  E  27,  Ash  20  N  29  W  31. 

In  a  similar  manner  the  quarter-section  corners  on  the  lines  H 
X,  I  W,  J  V,  etc.,  are  also  numbered  and  the  witnesses  given  to 
each. 

Each  of  these  lists  is  extended  so  as  to  include  all  the  corners  of 
that  particular  class. 

(  147  ).  The  following  particulars  are  usually  shown  on  the  face 
of  the  plot : 

Length  of  fractional  lines : 

B  to  6,  39.07. 

C  to  6,  38.49. 

D  to  6,  38.03,  and  so  on  with  all  the  other  fractional  lines. 


82  MANUAL  OF  PLANE  SURVEYING. 

1.  Area  of  fractional  quarters. 

N.  E.  qr.  sec.  1,  159.17  acres. 
N.  W.  qr.  sec.  1,  157.51  acres. 
N.  E.  qr.  sec.  2,  155.70  acres. 

2.  Creeks,  etc. 

N  from  R  31.42,  creek  running  S.  W.,  43  links  wide. 

E  from  25, 26, 35, 36;  22.16,  creek  running  S.W.,  41  links  wide. 

3.  Offsets. 

B  41  links  E.  of  corner  in  town  north. 
L  59  links  S.  of  corner  in  range  west. 

Only  a  few  instances  are  cited  in  each  case  to  show  the  general 
plan. 

SUBSEQUENT  NOTES. 

( 148  ).  Every  surveyor  should  be  provided  with  a  copy  of  the 
original  field-notes  of  his  county,  together  with  notes  of  all  the 
surveys  made  by  his  predecessors.  These  notes,  with  the  addi- 
tions he  himself  makes  from  time  to  time  (provided  he  and  his 
predecessors  are  authorized  surveyors),  constitute  the  surveyor's 
records  of  the  county. 

(  149  ).  These  records  should  contain  a  plot  of  each  piece  of 
land  surveyed,  showing  its  area  and  the  course  and  distance  of 
each  of  its  boundary  lines.  The  manner  of  drawing  these  plots  is 
explained  in  the  chapter  on  "  PLOTTING." 

(  150  ).  From  the  regular  county  record  is  usually  drawn  a 
pocket  record  for  field  use  (1)  of  the  original  survey,  and  (2)  of  the 
subsequent  surveys.  The  notes  of  the  original  survey  generally 
fill  but  a  small  book,  and  may  be  arranged  according  to  the 
method  given  above ;  but,  unless  the  county  is  unusually  small,  it 
is  more  convenient  to  have  a  separate  book  for  each  range  in 
which  to  enter  the  notes  of  the  subsequent  surveys. 

( 151 ).  Each  of  these  books  should  contain  at  least  twice  as 
many  pages  as  there  are  sections  in  the  range.  The  left-hand 
page  in  each  one  sh6uld  contain  a  plot  of  a  section  about  4  inches 
square  divided  into  quarters,  and  the  right-hand  or  opposite  page 
should  be  left  blank,  so  that  notes  of  the  successive  surveys  in  the 


MANUAL   OF  PLANE  SURVEYING. 


section  may  be  entered  upon  it.     These  refer  to  the  plot  by  num- 
bers or  letters  in  the  manner  shown  in  the  figure. 

1.  (Left-hand  page). 

SECTION , 

TOWN ,  RANGE 

t C  I 

k 


tO.  20 

^                                            * 

A 

^                                           $ 

75 

±±. 

Oa                 tO   .     18               *> 

ti                                        ^ 

h 

3                " 

A 

to  .   /f 

> 

£     40.  S/ 

g 

FIG.  21. 

(This  plot  is  only  one-fourth  the  size  suggested  above,  but  will 
serve  to  illustrate.) 

2.  (Eight  hand  page). 
a.  Be.  16  N  12  E  19,  Ash  14  S  6  E  12. 
B.  Pop.  28  S  60  W  19,  Walnut  30  N  16  E  41. 

c.  Be.  13  S  16  E  17,  Elm  18  N  22  E  31. 

d.  Pop.  16  N  19  E  27,  Ash  18  N  23  W  21. 
A.  Be.  23  S  17  W  40,  Elm  16  N  21  W  14. ' 

Bearings  of  principal  lines : 
a  f.   N  89°  357  E. 
c  e.   N    1°  16X  W. 

( 152  ).  In  these  cases  the  bearings  are  all  given  on  the  basis  of 
the  true  meridian.  When  the  magnetic  bearings  are  given,  they 
should  be  accompanied  with  the  date  at  which  they  were  taken. 


84  MANUAL   OF   PLANE   SURVEYING. 

( 153 ).  All  interior  surveys  in  the  quarter-sections  should  be 
represented  by  proper  lines  on  the  plot.  The  dotted  line  through 
the  north-east  quarter  in  the  figure  indicates  that  this  quarter  has 
been  divided  into  north  and  south  halves.  The  courses  of  roads 
and  creeks  may  also  be  shown  on  the  plot. 

(154).  When  the  bearing  of  any  independent  line  is  not 
known,  it  may  generally  be  approximated  by  comparing  it  with 
other  lines  in  the  section  or  adjoining  sections.  For  instance, 
there  is  but  little  difference  between  the  bearing  of  the  line  a  b 
in  the  figure  and  that  of  the  line  c  d,  since  the  distance  between 
their  northern  extremities  differs  but  4  links  from  the  distance  be- 
tween their  southern  extremities,  and  both  lines  are  about  of  the 
same  length. 

(155).  The  bearing  of  the  line  ab  may  be  determined  by  the 
following  method,  which  has  been  explained  in  a  preceding  chap- 
ter : 

70 : 4 : :  60/^z=3'-f-  =  amount  of  correction. 
1°  16'—  3/=l°  13'=  bearing  of  line  a  b. 

(156).  This  method  of  determining  the  bearing  of  one  line 
from  that  of  another  depends  on  the  following  principles:  (1) 
Two  parallel  lines  have  the  same  bearing,  and  (2)  the  difference 
in  bearing  of  two  lines  not  parallel  is  in  proportion  to  their  incli- 
nation to  one  another.  The  error  caused  by  convergence  in  north 
and  south  lines  may  be  disregarded  when  the  lines  are  short. 

By  reversing  this  rule  we  may  determine  the  distance  between 
two  lines  at  successive  points  when  their  distance  apart  at  one 
point  and  their  difference  of  bearing  are  known. 

(  157  ).  The  surveyor's  field-book  is  the  memorandum  he  keeps 
of  his  field-work.  It  contains  only  the  rough  entries,  which  are 
changed  in  form  and  transmitted  to  the  records. 

( 158  ).  In  surveys  of  independent  tracts,  perhaps  the  following 
method  of  keeping  it  is  as  good  as  any : 

Let  us  suppose  a  survey  of  the  south-east  quarter  of  section  9, 
town  8,  range  10,  commencing  at  the  south-east  corner  of  the  sec- 
tion, and  made  March  29,  1881.  If  all  the  corners  to  the  quarter- 
section  have  been  established  previously,  and  the  surveyor  runs 
the  east  line  first,  the  entries  may  be  somewhat  as  follows : 

Mar.  9,  1881. 
9  —  8  —  10. 

Commenced  S  E  cor.  and  ran  N  2°  15'  W,  40.20  to  E  }  cor.,  Be 


MANUAL    OF    PLANE   SURVEYING. 


85 


20  N  15  E  36,  Ash  10  N  41  W  12,  M  E  16  links.  At  16.30  from 
S  E  cor.  crossed  brook  8  links  wide  flowing  S  E. 

Com.  E  J  ran  S  88°  40'  W,  40.12  to  center  of  section.  19.00 
crossed  brook  6  links  flowing  S  E. 

Com.  cen.  ran  S  2°  30'  E,  40.20  to  S  J. 

Com.  S  J  ran  N  88°  40'  E,  40.15,  M  L  5  links. 

The  first  line  terminated  16  links  east  of  the  corner,  showing 
that  the  assumed  bearing  was  about  }  degree  too  small.  The 
second  and  third  lines  struck  the  corners,  but  the  fourth  ran  5 
links  to  the  north,  perhaps  on  account  of  some  slight  error  in  set- 
ting the  compass,  as  the  assumed  bearing  was  correct,  if  we  com- 
pare with  the  second  line  run. 

(159  ).  Wherever  the  witnesses  are  found  in  bad  condition  new 
ones  are  taken,  as  was  done  at  the  E  £  corner  in  this  survey. 

(160).  In  dependent  surveys  it  is  generally  best  to  have  the 
page  of  the  field-book  ruled  in  five  vertical  columns:  The  first 
giving  the  relative  name  or  number  of  the  station  or  corner  at 
which  the  line  begins;  the  second,  its  course;  the  third,  its  dis- 
tance; the  fourth,  the  number  of  links  missed  to  the  right;  the 
fifth,  the  number  of  links  missed  to  the  left,  as  follows : 

Sec.  5,  Town  6,  Kange  4. 
Mar.  18,  1881. 


Sta. 

Course. 

Dis. 

R. 

L. 

A 

N10°W 

16.00 

4 

B 

N   4°W 

4.00 

C 

N  16°  3<y  E 

18.00 

6 

D 

ssi°iyvf 

9.00 

E 

N  14°  W 

11.24 

F 

S29°W 

7.26 

5 

FIG.  22. 

( 161 ).  The  station  at  which  the  surveyor  begins  is  called  "A," 
and  the  succeeding  ones  are  named  in  the  order  of  the  letters  that 
follow. 

Witnesses  taken  at  any  of  the  corners  may  be  described  on  the 


86  MANUAL  OF   PLANE  SURVEYING. 

opposite  page  and  referred  to  the  corner  by  the  proper  letter,  n» 
follows : 

At  B.   Be  24  N  23  W  16,  S  E  corner  house  N  81  W  46. 
"  E.   Elm  22  S  16  E  32,  Oak  23  M  12  W  29. 
"  F.    Large  stone  at  corner. 

The  names  of  all  the  assistants  in  the  survey  are  generally  re- 
corded, so  that  they  may  be  known,  if  evidence  should  be  needed* 
in  case  of  future  disputes  over  the  survey. 

( 162).  The  record  made  by  county  and  other  authorized  sur- 
"veyers  is  taken  as  prima  facie  evidence  in  favor  of  the  surveys 
made  by  them,  and  particular  care  should  be  taken  in  making 
this  record,  as  well  as  in  field-work,  to  see  that  no  mistakes  are 
committed. 

( 163 ).  Other  methods  of  keeping  field-notes  are  also  employed 
by  surveyors,  but  the  ones  described  above  are  perhaps  the  most 
simple,  and  they  will  answer  every  purpose. 

QUESTIONS  ON  CHAPTER  VIII. 

1.  What  particulars   are  enumerated  in   the  original   field- 

notes? 

2.  By  referring  to  the  original  notes,  how  would  you  find  the 

witness  to  the  south-west  corner  of  section  21  ?     The  W  £ 
corner  of  section  9?    The  S  J  of  section  22? 

3.  What  sections  touch  section  15?     29?     26? 

4.  What  do  you  understand  by  "Subsequent  Notes?" 

5.  What  constitute  the  surveyor's  records  of  a  county? 

6.  How  is  the  bearing  of  an  independent  line  sometimes  ap- 

proximated? 

7.  The  north  line  of  a  quarter-section  is  40.32  long,  and  the 

south  line  39.97  long.     If  the  bearing  of  the  east  line  is 
N  2°  19  W,  what  is  the  bearing  of  the  west  line? 

8.  Describe  the  field-book   for  independent   tracts.     For  de- 

pendent tracts. 

9.  When  are  new  witnesses  taken  to  a  corner? 

10.  What  is  meant  by  prima  facie  evidence?  Answer — Evidence 
that  is  held  to  be  good  until  set  aside  by  stronger  evi- 
dence. 


CHAPTER  IX. 

RE-LOCATION  OF  CORNERS. 

( 164).  Perhaps,  as  a  general  thing,  the  most  perplexing  part 
of  a  surveyor's  work  consists  in  re-locating  the  corners  of  the 
original  survey.  As  long  as  these  or  the  witnesses  to  them  re- 
main, he  seldom  has  any  serious  trouble  in  the  survey  of  any 
independent  line  of  the  section;  but  if  one  of  them  chance  to 
he  lost,  it  must  be  re-located  before  any  line  bearing  on  it  can  be 
surveyed,  and  its  re-location,  except  in  certain  cases,  is  frequently 
a  matter  of  difficulty,  if  not,  in  some  instances,  of  impossibility. 

In  the  latter  case  there  is  no  alternative,  except  to  establish  a 
new  corner. 

(165).  This  difficulty  arises  from  the  fact  before  stated,  that 
section  lines  are  usually  broken  at  every  original  corner,  and 
these  parts,  into  which  the  lines  are  divided,  differ  from  one  an- 
other in  length,  so  that  the  course  and  distance  of  one  line  may 
not  be  the  same  as  that  of  any  other  line  in  the  vicinity,  and  it 
would  not  do,  therefore,  to  re-locate  a  corner  by  a  line  having  the 
same  course  and  distance  of  a  similar  line,  either  in  the  same  sec- 
tion or  any  other  section.  The  surveyor,  consequently,  must 
resort  to  other  means. 

1.  In  the  first  place  a  diligent  search  should  be  made  for  re- 
mains of  the  monument  or  witnesses  at  the  missing  corner.     If 
the  witness  trees  have  disappeared,  it  may  be  possible  to  find  the 
roots,  especially  if  the  ground  has  not  been  plowed,  as  traces  of 
them  remain  for  many  years.     It  is  probable  that  some  person  in 
the  vicinity  can  give  him  some  information  relative  to  the  corner 
that  will  enable  him   to   judge  approximately  as   to  where  he 
should  look  for  the  witnesses. 

2.  If,  however,  all  search  prove  futile,  and  the  course  and  dis- 

(87) 


MANUAL   OP   PLANE  SURVEYING. 

tance  of  a  line  connecting  this  corner  with  some  other  corner  that 
can  be  found,  be  known,  the  missing  corner  may  sometimes  be 
found  by  running  this  line  from  the  known  corner. 

3.  Or  if  the  line  run  through  the  woods,  it  may  be  possible  to 
retrace  it  by  the  blazes  on  the  trees,  and  thus  determine  the  miss- 
ing corner,  providing  the  length  of  the  line  be  known. 

4.  And,  again,  where  subsequent  surveys  have  been  made  in 
one  of  the  sections  touching  the  corner,  some  of  the  subsequent 
corners,  taken  in  connection  with  the  original  corners,  may  enable 
the  surveyor  to  re-locate  the  missing  corner  by  "projection,"  a* 
follows : 

(1).  Suppose  the  S  J  corner  of  the  section  to  be  missing,  and 
that  the  S  E  corner  and  S  £  }  E  corner  can  both  be  found.  Now, 
the  S  £  J  E  corner  was  evidently  set  while  traces  of  the  S  £  corner 
remained,  and  is  midway  between  it  and  the  S  E  corner.  If  now 
we  begin  at  the  S  E  corner  and  survey  a  line  westward,  measuring 
to  the  S  J  J-  E  corner  and  producing  the  line  an  equal  distance 
beyond  it,  the  extremity  of  this  line  must  mark  the  missing  cor- 
ner. If  the  line  run  either  to  one  side  or  the  other  of  the  S  J  \  E 
corner,  the  distance  to  the  right  or  left  must  be  noted.  The  S  \ 
corner  will  be  twice  this  distance  on  the  same  side  from  the  ex- 
tremity of  the  line,  as  may  be  seen  by  noticing  Fig.  23. 

The  numbers  on  the  horizontal  (true)  line  indicate  the  length 
of  each  section  of  it,  and  those  on  the  dotted  vertical  lines  show 
the  distance  between  the  random  line  and  the  true  line  at  each  of 
the  corners. 

(2).  This  is  simply  the  reverse  of  the  method  used  in  setting 
the  S  \  \  E  corner. 

(3).  If,  instead  of  the  S  E  corner  and  S  \  \  E  corner,  we  have, 
for  instance,  the  S  E,  E  \  },  and  S  E  center  corners,  it  will  be 
necessary  first  to  project  the  S  \  \  E  from  the  E  \\  and  S  E  cen- 
ter, and  then  we  may  project  the  S  £,  same  as  before. 

EXAMPLES  : 

(4).  1.  If  we  have  the  center  and  S  \  \  corners,  how  may  we 
find  the  S  i? 

2.  The  W  J,  S  \  },  and  S  W  center  corners  are  known,  how  may 
the  S  W  corner  be  determined? 

3.  The  E  },  E  \  },  and  S  \  },  corners  can  be  found,  how  may  the 
S  \  be  re-located? 


MANUAL   OF   PLANE  SURVEYING. 


(5).  It  must  be  borne  in  mind  that  two  corners  must  be  found 
on  the  same  side  of  a  quarter-section  before  the  third  can  be  re- 
located by  this  method. 


16 


FIG.  23. 

5.  When  the  surveyor  finds  it  impossible  to  re-locate  a  missing 
corner  by  any  of  the  foregoing  methods,  or  any  other  method,  he 
proceeds  to  establish  a  new  corner,  and  in  doing  this  he  presumes 
that  the  quarter-section  lines  do  not  bend  at  the  corner  to  be  established, 
and,  if  it  be  a  quarter-section  corner,  that  it  is  midway  between  the 
corners  of  the  section. 

NOTE  ON  SECTION  5,  ARTiCLE(165).— The  surveyor  must  bear  in  mind  that  the 
method  here  proposed  is  to  be  used  only  as  a  last  resort.  The  original  field 
notes  nearly  always  give  the  length,  and  sometimes  the  bearing,  of  every 
half-mile  of  line  surveyed,  and  by  foilo\ying  these  as  nearly  as  possible  and 
noticing  the  recorded  objects  on  the  line,  the  missing  corner  may  gener- 
ally be  re-located  without  resorting  to  this  method.  In  Fig.  24  the  cor- 
ner might  be  set  by  taking  the  distances  on  the  two  half  mile  lines,  as 
shown  by  the  original  notes,  and  in  Fig.  25  by  taking  the  distances  on  the 
four  half-mile  lines  as  shown  by  the  same.  If  the  measurements  do  not 
correspond  with  those  of  the  Government  Deputy,  proportionate  distances 
must  be  taken  and  the  corner  set  by  them.  This  is  the  authorized  method, 
and  the  principles  involved  are  general  in  their  application. 


90  MANUAL   OF   PLANE  SURVEYING. 

( 1 ).  Suppose  the  missing  corner  to  be  the  S  J,  the  new  corner 
would  be  set  by  bisecting  the  line  connecting  the  S  E  and  S  W  cor- 
ners of  the  section  in  the  same  way  that  a  i  }  corner  is  set  by  bi- 
secting the  line  between  the  two  corners  of  the  quarter-section. 

(  2  ).  The  corner  thus  established  may  be  identical  with  the  lost 
corner,  or  may  be  some  distance  from  it,  but  it  is  the  best  that  can 
be  done. 

(o).   In  the  figure  a  case  is  illustrated  in  which  the  new  corner 


f 


40,21          §>/4  40.21 

FIG.  24. 

is  a  considerable  distance  from  A.  the  supposed  approximate  loca- 
tion of  the  old  corner.  The  dotted  lines  represent  the  old  lines, 
and  the  numbers  below  the  new  line  show  the  length  of  each  sec- 
tion of  it. 

(4).  If  the  corner  of  a  section  be  lost,  a  new  one  is  set  by  sur- 
veying the  exterior  lines  of  the  adjacent  quarter-sections  as  if  they 
did  not  bend  at  the  section  corner.  The  new  corner  will  be  at  the 
point  of  intersection  of  the  two  lines  thus  surveyed. 

For  instance,  to  set  a  new  S  E  corner  to  section  2,  connect  the 
S  J  corner  of  section  1  with  the  S  ^  corner  of  section  2,  and  the  E 
J  corner  of  section  2  with  the  E  }  corner  of  section  11.  The  point 
at  which  the  lines  cross  will  be  the  new  corner. 

(5).  As  in  the  former  case,  this  corner  may  not  be  identical 
with  the  old  corner.  Fig.  25  represents  a  possible  case  in  which 
they  are  some  distance  apart.  In  actual  work,  however,  such  ex- 
treme cases  as  we  have  noticed  will  seldom,  if  ever,  come  up. 

(  6 ).  In  any  case,  in  setting  a  new  section  corner,  if  any  J  cor- 
ner can  not  be  found,  the  line  must  be  produced  to  the  next  corner 
that  can  be  found.  This  may  cause  one  of  the  lines  to  be  li  or 
even  2  miles  long,  but  the  corner  is  set  at  the  point  of  intersection, 
same  as  before. 

(166  ).   In  re-locating  the  original  corners  to  the  variable  quar- 


MANUAL   OF   PLANE   SURVEYING. 

ters  of  fractional  sections  any  of  the  first  four  methods  given  above 
may  be  employed,  but  when  a  new  corner  must  be  established  the 
oth  or  last  does  not  always  hold  good,  except  for  the  \  corner  on 
the  town  or  range  line,  which  is  set  midway  between  the  section 
corners,  as  in  full  sections. 

( 167 ).   To  set  the  i  corner  between  two  fractional  sections,  run 


Set.    Z.     / 


'#     cor, 


Sec,  lt 


Sec,   II. 


^      COT. 

FIG.  25. 

a  line  from  the  interior  section  corner  westward  or  northward,  as 
the  case  maybe,  to  the  exterior  section  corner  on  the  town  or  range 
line,  and  locate  the  corner  40  chains  from  the  starting  point. 

( 168 ).  To  set  an  exterior  corner  to  a  fractional  section,  or  to 
any  exterior  section. 

1.  If  there  be  an  offset  between  the  corner  of  one  section  and 
that  of  the  corresponding  section  in  the  other  town  or  range,  the 


92  MANUAL   OF  PLANE   SURVEYING. 

corner  may  be  re-located  by  measuring  this  offset  along  the  town 
or  range  line,  or  correction  parallel,  in  the  proper  direction. 

2.  When  there  is  no  offset,  the  corner  must  be  set  by  crossing 
the  lines,  according  to  the  method  used  in  interior  sections. 

EXAMPLES. 

( 169  ).  1.  How  may  a  new  E  }  corner  be  set  to  section  11,  pro- 
viding the  section  corners  on  that  side  can  both  be  found? 

2.  What  must  be  done,  if  one  or  both  of  the  section  corners  are 
lost,  before  the  quarter-section  corner  between  them  can  be  set? 

3.  How  do  you  establish  a  new  interior  section  corner? 

4.  If  one  or  more  of  the  exterior  corners  to  the  adjacent  quar- 
ter-sections be  lost,  what  must  be  done  in  order  to  establish  the 
section  corner? 

5.  How  do  you  establish  the  ^  corner  section  between  two  frac- 
tional sections  ? 

6.  How  do  you  re-locate  an  exterior  section  corner  when  there 
is  an  offset  on  the  town  or  range  line? 

(170).  When  possible,  all  corners,  whether  independent  or  de- 
pendent, are  re-located  according  to  the  rules  by  which  they  Vrere 
located  at  first. 

QUESTIONS  ON  CHAPTER  IX. 

1.  When  it  is  found  impossible  to  re-locate  a  corner,  what  must 

be  done?  [corner? 

2.  Why  is  not  the  new  corner  always  identical  with  the  original 

3.  Explain  each  of  the  different  methods  of  re-locating  original 

corners. 

4.  In  re-locating  an  original  J  corner  by  "projection,"  the  ran- 

dom line  was  found  to  be  16  links  to  the  left  of  the  $  ^ 
corner.  In  what  direction  and  how  far  will  the  re-located 
corner  be  from  the  extremity  of  the  random  line? 

5.  How  many  corners  must  be  found  on  the  side  of  a  quarter- 

section  before  the  remaining  one  can  be  re-located  ?    Why  ? 

6.  What  section  in  the  township  north  corresponds  with  section 

2?  With  section  5?  What  one  in  the  township  west  cor- 
responds with  7?  With  30? 

7.  What  townships  touch  T  2  N,  R  3  E?    T4S,  B5W?    Tft 

N,  E  1  W? 

8.  How  are  subsequent  corners  re-located? 


CHAPTER    X. 

DESCRIPTIONS  OF  LAND. 

(171 ).  No  piece  of  land  can  be  sold  or  surveyed,  unless  its  de- 
scription is  known,  and  this  description  should  be  just  as  concise 
and  simple  as  possible. 

(172).  It  would  be  better,  if  the  length  of  lines  and  area  of 
tracts  were  always  given  in  surveyor's  measure,  instead  of  in  or- 
dinary linear  and  square  measure;  yet  when  this  is  not  done,  they 
may  be  reduced  to  their  equivalents  in  surveyor's  measure  by  the 
following  tables : 

(173).  LINEAR  MEASURE. 

100  links  =    1  chain  =  4  rods. 
1,000    "      =10      "      =1  furlong. 
8,000     "      =80      "      =1  mile. 

SQUARE  MEASURE. 
1  sq.  chain  =  16  sq.  rods. 
10   "       "       =1  acre. 
6,400  "      "      =    Isq.  mile. 

It  is  plain  that  rods  maybe  reduced  to  chains  by  dividing  by  4; 
furlongs  to  chains  by  multiplying  by  10 ;  and  miles  to  chains  by 
multiplying  by  80. 

EXAMPLES. 

(174).    1.   Reduce  15  rods  to  chains. 

2.  Reduce    1  fur.  3  rods  to  chains. 

3.  Reduce    1  mi.  3  fur.  24  rods  to  chains. 

(175).  Fractional  parts  of  a  chain  should  be  expressed  in 
links:  Thus,  lOf  chains  should  be  written  10  chains  and  seventy- 
five  links,  or  simply  10.  75. 

(93) 


94  MANUAL   OF   PLANE  SURVEYING. 

(176).  As  a  general  :mle  for  reducing  from  ordinary  long  or 
linear  measure  to  surveyor's  measure,  perhaps  it  would  be  well  to 
use  the  following: 

Eeduce  the  denominations  expressing  the  length  of  the  line  to 
rods,  and  multiply  by  .25.  The  product  will  be  the  length  of  the 
line  expressed  in  chains  and  links. 

To  illustrate,  suppose  the  length  of  a  line  to  be  7  fur.  16£  rods  = 
296J  rods.  =  296.5  rods.  This  multiplied  by  .25  equals  74.125,  or 
74  chains  12J  links. 

(177).  The  area  of  tracts  in  surveyor's  measure  is  always 
given  in  acres  and  hundredths,  instead  of  in  acres,  roods,  rods, 
etc.,  as  ordinarily.  This  will  be  explained  in  the  chapter  on 
Computation  of  Area. 

(178).  Whenever  an  independent  tract  is  to  be  described, 
nothing  whatever  should  be  said  of  the  course  and  distance  of  any 
of  its  boundary  lines,  and  it  should  be  described  simply  as  such  a 
division  or  sub-division  of  the  section ;  as,  for  instance,  the  south- 
west quarter,  or  the  north  half  of  the  north-east  quarter,  or  the 
north-west  fourth  of  the  south-east  quarter,  or  the  north  half  of 
the  south-east  fourth  of  the  north-west  quarter,  etc.,  etc. 

(  179).  Errors  like  the  following  are  frequently  made  in  de- 
scriptions: Forty  acres  in  the  form  of  a  square  in  the  south-east 
corner  of  the  section ;  eighty  acres  off  the  south  side  of  the  north- 
east quarter;  one  hundred  and  sixty  acres  in  the  north-east  corner 
of  the  section ;  a  strip  twenty  chains  wide  off  the  north  side  of  the 
south-west  quarter;  and  so  on. 

Each  of  these  descriptions  is  faulty,  because  the  independent 
division  intended  to  be  described  may  overrun  or  fall  short  in  the 
amount  of  land  named  in  the  description.  If  the  tracts  were  not 
independent,  the  descriptions  would  be  good. 

Correct  the  following  descriptions  : 

( 180 ).   1.   160  acres  off  west  side  of  section. 

2.  Forty  acres  in  the  south-west  corner  of  the  north-east  quarter. 

3.  Commencing  at  the  X  E  cor.  of  the  section  ;  thence  running 
south  20  chains;  thence  west  40  chains;  thence  north  20 chains  to 
the  N  }  cor.;  thence  east  to  the  place  of  beginning.     Containing 
80  acres. 

4.  60  acres  off  the  south  side  of  the  south-east  quarter. 


MANUAL   OF   PLANK   SURVEYING.  95- 

(181 ).  Sometimes  descriptions  contain  errors  that  render  them 
worthless.  The  following  are  a  few  examples;  tell  where  the 
error  lies  in  each  one : 

1.  NEJNWqr. 

2.  S  W  }  N  E  qr.,  containing  80  acres. 

3.  SJNJXEqr. 

4.  60  acres  in  N  E  qr. 

5.  N  W  qr  sec.  28,  containing  80  acres. 

6.  Running  north ;  thence  east  50  chains. 

7.  Eunning  S  43°  E,  11.21 ;  thence  N  32°  W,  5.26 ;  thence 

S  8°  3V  E,  16.32,  to  the  place  of  beginning. 

Mistakes  like  the  preceding  are  frequently  made  by  persons  who 
are  careless,  or  do  not  understand  how  lands  should  be  described> 
and  sometimes  give  rise  to  vexatious  litigation. 

( 182  ).  It  is  best  in  nearly  all  cases  to  qualify  the  area  of  the 
tract  described  by  the  phrase  "  more  or  less,"  as,  perhaps,  no  two 
surveys  of  the  same  tract,  particularly  if  it  be  large,  will  ex- 
actly coincide  throughout,  and  of  course  the  area  will  vary  with 
the  length  of  the  lines. 

( 183 ).  In  describing  dependent  tracts  the  course  and  distance 
of  each  of  their  boundaries  should  be  given,  except,  perhaps,  in 
occasional  cases  where  they  have  a  natural  or  artificial  boundary, 
as  for  instance,  a  creek  or  road,  whose  course  and  distance  may  be 
determined  at  any  time;  but  it  is  always  best  to  be  definite  in  re- 
gard to  boundaries  when  possible. 

( 184  ).  The  description  should  also  state  whether  the  bearings 
are  based  on  the  true  meridian  or  on  the  magnetic  meridian.  If 
based  on  the  magnetic  meridian,  the  date  at  which  they  were 
taken  should  be  given. 

( 185  ).  Where  a  line  is  described  as  running  north,  a  due  north 
and  south  line  is  meant,  and  the  same  is  true  of  south.  Similarly, 
an  east  line  means  one  running  due  east,  and  a  west  line  one 
running  due  west. 

(  186).  The  survey  of  a  tract  of  land  is  always  made  in  ac- 
cordance with  the  description,  except  where  an  obvious  mistake 
occurs,  in  which  case  the  surveyor  will  have  to  exercise  his  judg- 
ment in  regard  to  the  course  to  be  pursued,  as  no  rule  can  be  given 
that  will  apply  to  all  cases.  However,  the  decisions  in  the  "Ap- 
pendix "  may  assist  him  somewhat  in  arriving  at  a  conclusion. 


96  MANUAL  OF  PLANE  SURVEYING. 

Sometimes  the  mistake  is  made  in  writing  the  original  descrip- 
tion of  the  tract,  and  at  others  in  copying  from  preceding  titles 
and  deeds.  In  the  latter  case,  a  comparison  of  the  deeds  will  show 
in  what  it  consists.  As  soon  as  a  mistake  is  discovered  in  a  deed 
or  mortgage,  or  in  any  other  instrument  in  which  a  great  deal 
may  depend  upon  the  description,  steps  should  be  taken  by  those 
interested  to  have  it  corrected. 

«     None  but  competent  persons  should  be  chosen  to  write  descrip- 
tions of  land. 

QUESTIONS  ON  CHAPTER  X. 

1.  Why  can  not  a  tract  of  land  be  sold  or  surveyed  without  a 
description  ? 

2.  How  many  links  in  a  rod?     Chains  in  a  mile? 

3.  Write  17  chains  and  46^  links  decimally. 

4.  Give  the  general  rule  for  reducing   from  ordinary  long  or 

linear  measure  to  surveyor's  measure. 

5.  How  is  the  area  of  a  tract  of  land  expressed  in  surveyor's 

measure? 

6.  Why  should  not  the  metes  and  bounds  of  an  independent 

tract  be  given  in  a  description  ? 

7.  Why  should  the  phrase  "  more  or  less  "  be  inserted  in  a  de- 

scription ? 

8.  Why  is  it  necessary  to  state  whether  the  bearings  are  based 

on  the  true  meridian  or  on  the  magnetic  meridian? 

9.  If  on  the  magnetic  meridian,  why  should  the  date  at  which 

they  were  taken  be  given  ? 


CHAPTER    XI. 

OBSTACLES  TO  ALIGNMENT  AND  MEASUREMENT. 

( 187  ).  It  frequently  happens  in  the  course  of  a  survey  that  the 
line  strikes  an  obstacle  of  some  kind — as,  for  instance,  a  building, 
or  a  large  pond,  or  creek — that  obstructs  the  measurement,  if  not 
both  line  of  sight  and  measurement. 

(188).  These  obstacles  may  be  divided  into  two  classes:  (1)  Ob- 
stacles that  may  be  spanned  by  measurements  along  their  sides 
or  margins,  as  a  building,  a  pond,  etc. ;  (2)  obstacles  that  can  not 
be  spanned  in  this  way,  as  rivers  and  lakes. 

( 189).  Various  methods  are  employed  for  spanning  obstacles, 
but  only  a  few  will  be  given,  in  order  to  prevent  confusion. 

FIRST  CLASS  OF  OBSTACLES. 
1.  By  Perpendiculars.— Fig.  26  represents  an  obstacle  on  the  line 


FIG.  26. 

A  B  which  runs  nearly  to  the  side  of  it.  At  the  extremity  B,  a 
perpendicular,  B  C,  is  measured  long  enough  to  permit  the  line  C 
D  to  pass  the  obstacle.  In  this  case  the  perpendicular  is  fifty  link* 
long.  The  line  C  D  is  then  run  at  the  bearing  of  the  line  A  B,  and 
is,  consequently,  parallel  to  it.  From  the  extremity,  D,  of  this 
line  another  perpendicular,  D  E,  of  the  same  length  as  the  first, 
7  (97) 


98  MANUAL   OF   PLANE  SURVEYING. 

is  measured,  which,  of  course,  terminates  on  the  original  line  pro- 
duced through  the  obstacle.  The  survey  of  the  line  may  then  be 
continued  from  E  in  the  direction  E  F  at  pleasure,  and  the  length 
of  C  D  added  to  the  regular  sections,  A  B  and  E  F. 

2.   By  an  Equilateral  Triangle. — The  line  A  B  terminates  some- 
what further  from  the  side  of  the  obstacle  than  before,  and  the 


FIG.  27. 

line  B  C  is  then  laid  off  at  an  angle  of  60°  with  the  line  A  B  pro- 
duced and  measured  to  a  suitable  distance.  In  the  case  before  us 
it  is  1  chain  and  25  links  in  length.  From  the  extremity,  C,  of 
this  line,  the  line  C  D,  of  equal  length  with  it,  is  surveyed  at  an 
angle  of  60°  with  C  B.  We  then  have  an  equilateral  triangle,  and 
the  side  B  D  is  also  1  chain  and  25  links  in  length.  The  line  may 
then  be  continued,  and  the  distance  through  the  obstacle,  1  chain 
and  25  links,  added  to  the  other  sections,  as  before. 

3.   (a)   By  a  Right-angled  Triangle. — This  method  is  similar  to 
the  preceding  one,  and  differs  from  it  only  in  having  a  right-angle 


FIG.  28. 

at  C,  and  angles  of  45°  at  B  and  D  in  the  triangle  used.    The  side 
B  C  is  first  surveyed,  and  then  C  D  at  right-angles  to  it  and  of 


MANUAL   OF   PLANE   SURVEYING. 

equal  length.  The  distance  from  B  to  D  is  found  by  extracting 
the  square  root  of  the  sum  of  the  squares  of  B  C  and  C  D,  as  the 
side  B  D  is  the  hypothenuse  of  the  triangle.  In  this  case  the  dis- 
tance from  B  to  D  is  equal  to  y(100)»+  (100)*  =  1.414. 

(6)  When  the  obstacle  is  a  pond,  or  something  that  does  not 
obstruct  the  line  of  sight,  the  following  method  will  be  found  most 
convenient : 

C 


FIG.  29. 

The  line  is  measured  to  B,  near  the  margin  of  the  pond,  and  the 
flag  set  at  D  on  its  continuation  on  the  opposite  side.  C  B  is  then 
measured  perpendicular  to  A  B,  and  lastly  the  line  C  D  is  meas- 
ured. We  have  now  a  right-angled  triangle  whose  base  is  required 
and  may  be  found  by  extracting  the  square  root  of  the  difference 
between  the  squares  of  C  D  and  B  C.  In  this  example  the  base 
B  D  equals  /(125)2  —  (75)2  =  1.00. 

In  every  case  the  line  is  to  be  continued  from  D  at  the  bearing 
of  the  first  section,  A  B,  and  the  distance  through  the  obstacle 
must  be  added. 

4.  By  Symmetrical  Triangles.— When,  as  in  the  last  case,  the  line 
of  sight  is  not  obstructed,  the  following  method  may  sometimes 
be  used : 

F 


A 


FIG.  30. 


100  MANUAL  OP  PLANE  SURVEYING. 

From  the  extremity,  B,  of  the  line  A  B  measure  a  line  to  C  and 
produce  it  to  F,  an  equal  distance  beyond,  and  then  from  D  meas- 
ure the  line  D  E  so  that  C  will  be  in  the  center.  The  line  E  F 
will  then  be  equal  to  the  line  B  D. 

5.  When  a  fence  is  built  on  a  line  to  be  surveyed,  it  is  best  to 
take  an  offset  either  to  one  side  or  the  other,  and  allow  for  it  when 
the  stakes  are  set  on  the  true  line,  or  the  stakes  may  be  moved  back 
a  distance  equal  to  the  offset  as  they  are  set.  They  will  thus  be  on 
the  random  line,  and  may  be  corrected  the  same  as  if  no  offset 
had  been  taken. 

It  is  customary,  after  an  offset  has  been  taken,  to  measure  back 
to  the  random  line  as  soon  as  the  obstruction  is  cleared,  but  if  the 
corner  be  reached  before  this  is  done,  the  offset  must  not  be  forgot- 
ten in  measuring  the  distance  the  line  runs  to  the  right  or  left  of  it. 

In  doing  this  observe  the  following  rules  : 

(1).  When  the  offset  is  taken  either  to  the  right  or  left  and  the 
offset  line  terminates  on  the  opposite  side  of  the  corner,  the  dis- 
tance missed  by  the  random  line  will  be  equal  to  the  distance 
missed  by  the  offset  line,  plus  the  offset,  and  it  will  terminate  on 
the  same  side  of  the  corner  as  the  offset  line. 


A. 
IQ\ 


FIG.  31. 


In  the  figure  an  offset,  A  B,  of  10  links  was  taken  to  the  right, 
and  the  offset  line,  B  C,  ran  10  links  to  the  left  of  the  corner  A. 
The  random  line,  A  B,  will  therefore  terminate  20  links  to  the  left 
of  the  corner. 

(2).  When  the  offset  line  terminates  on  the  same  side  as  that  on 
which  the  offset  is  taken. 

This  involves  two  cases:  (a)  When  the  distance  missed  is 
greater  than  the  offset,  and  (6)  when  the  offset  is  greater  than  the 
distance  missed. 

(a).  Subtract  the  offset  from  the  distance  that  the  offset  line 
misses  the  corner;  the  remainder  will  be  the  distance  missed  by 
the  random  line.  The  termination  of  the  random  line  will  lie  on 
the  same  side  of  the  corner  as  the  termination  of  the  offset  line. 

(6).   Subtract  the  distance  missed  by  the  offset  line  from  the 


MANUAL   OF   PLANE  SURVEYING.  101 

offset ;  the  difference  will  equal  the  distance  missed  by  the  random 
line.  The  termination  of  the  random  line  will  be  on  the  opposite 
side  of  the  corner  from  the  termination  of  the  offset  line. 

The  offset  line  is  always  parallel  to  the  random  line. 

In  correcting  the  stakes  on  the  offset  line,  it  is  best  to  correct  as 
if  they  were  on  the  random  line,  and  then  move  them  a  distance 
equal  to  the  size  of  the  offset,  and  in  a  direction  opposite  to  that 
in  which  the  offset  is  taken.  This  will  put  them  on  the  true  line. 

6.  Sometimes,  when  surveys  are  made  over  hills,  it  is  impossi- 
ble for  the  chain-men  to  see  the  compass  or  flag  to  which  they  are 
running.  In  this  case  a  stake  should  be  put  up  at  some  promi- 
nent point  on  the  line  by  the  surveyor,  to  which  they  may  meas- 


FIG.  32. 

ure  until  they  come  in  sight  of  the  compass  or  flag.  For  instance, 
if  the  chain-men  are  down  in  the  valley  A,  of  the  figure,  a  stake 
or  flag  should  be  set  on  the  ridge,  as  it  is  impossible  for  them  to 
see  the  compass  at  B. 

In  chaining  up  and  down  hill  it  is  frequently  necessary  to 
double  the  chain  or  divide  it  into  two  sections,  so  that  it  may  be 
held  in  a  horizontal  position.  A  light  steel  chain  is  always  pref- 
erable to  a  clumsy  iron  one,  as  it  will  not  sag  so  much. 

SECOND  CLASS  OF  OBSTACLES. 

(190).  1.  Suppose  the  obstacle  to  be  a  large  creek.  The  line 
is  surveyed  up  somewhere  near  the  edge  and  the  flag  set  on  the 
line  on  the  opposite  shore.  In  Fig.  33,  let  A  represent  the  point 
to  which  the  line  is  measured,  and  B  the  flag  set  on  the  opposite 
shore.  From  the  point  A,  a  line  of  indefinite  length  is  sighted  at 
right  angles  to  A  B.  The  compass  is  then  set  at  any  point  not  too 
near  A,  as  C,  on  this  line,  and  turned  so  the  sights  will  strike  B. 
The  size  of  the  angle  A  C  B  is  then  noted,  and  the  point  D  on  the 


102 


MANUAL   OF   PLANE   SURVEYING. 


line  A  E  sighted  at  an  equal  angle  on  the  other  side  of  A  C.     The 
distance  from  A  to  B  will  then  equal  the  distance  from  A  to  D. 


FIG.  33. 


2.   From  the  point  A,  a  perpendicular,  A  C,  may  be  sighted  and 
another,  C  D,  set  off  from  its  extremity.     The  point  E,  on  the  line 


FIG.  34. 
A  C,  is  then  found,  and  each  of  the  sections,  E  C  and  E  A,  meas- 


MANUAL   OF   PLANE  SURVEYING. 


103 


«red.     The  distance  from  A  to  B  may  then  be  found  by  the  fol- 
lowing proportion : 

CE    :    EA    ::    CD    :    (x  =  A  B) ; 
E  AXCD 


whence  A  B  = 


CE 


3.   A  perpendicular  is  set  off  from  the  line  B  F  at  F,  and  another 
at  A,  extended  to  the  line  B  D.     The  distances,  A  F,  A  C,  and 


D 


FIG.  35. 


whence  A  B  = 


D  F,  are  measured  and  the  distance  A  B,  found  as  follows : 
(D  F  —  A  C)    :    AC    :  :    A  F    :    (x  =  A  B) ; 
ACXAF 
'  —  AC 

(  191 ).   A  great  many  other  methods  might  easily  be  given,  but 
these  will  suffice. 

These  methods  will,  of  course,  answer  equally  well  where  the 
point  B  is  inaccessible  and  at  the  termination  of  a  line. 

In  field-work  the  method  that  seems  best  adapted  to  the  peculi- 
arities of  the  case  should  be  adopted. 

QUESTIONS  ON  CHAPTER  XI. 

1.  What  is  meant  by  an  obstacle  to  measurement?    To  align- 

ment? 

2.  How  many  classes  of  obstacles  are  there?    Name  one  of 

each  class. 


104  MANUAL   OF   PLANE  SURVEYING. 

3.  Describe  the  method  of  spanning  an  obstacle  by  perpen- 

diculars.    By  an  equilateral  triangle. 

4.  In  the  survey  of  a  certain  line  an  obstacle  is  met.     A  line 

is  then  surveyed  1  chain  and  40  links,  bearing  from  the 
termination  of  the  main  line  so  as  to  pass  the  obstacle. 
From  the  end  of  this  line  a  perpendicular  90  links  long 
is  measured  back  to  the  original  line  produced  through 
the  obstacle.  What  is  the  distance  through  the  obstacle? 

5.  Describe  the  method  by  symmetrical  triangles. 

6.  What  is  an  offset? 

7.  What  is  the  difference  between  the  offset  line  and  the  ran- 

dom line? 

8.  An  offset  of  12  links  is  taken  to  the  right,  and  the  offset 

line  misses  13  links  to  the  left.  How  far  will  the  random 
line  miss  the  corner? 

9.  What  two  cases  arise  when  the  offset  and  termination  of  the 

offset  line  both  lie  on  the  same  side  of  the  corner? 

10.  Give  the  rule  for  correcting  the  stakes  on  an  offset  line. 

11.  How  do  you  set  an  intermediate  stake  for  the  flag-men  to 

run  to  when  an  elevation  of  land  prevents  them  from  see- 
ing the  compass  or  flagstaff? 

12.  When  is  it  necessary  to  double  the  chain  or  divide  it  into 

two  sections  ? 

13.  Explain  each  of  the  methods  used  in  the  second  class  of 

obstacles. 

14.  In  Fig.  34,  A  E  =  1.40,  C  E  =  90,  and  C  D  =  1.10.    What 

is  the  length  of  A  B  ? 

15.  In  Fig.  35,  AC  =  1.22,  A  F  =  98,  and  DF=  1.54.    What 

is  the  length  of  A  B? 


CHAPTER  XII. 

COMPUTATION  OF  AREA. 

( 192 ).  In  computing  areas  the  length  of  all  lines  should  be 
expressed  in  chains,  chains  and  links,  or  links,  as  the  case  may 
be,  and  the  areas  may  then  be  reduced  to  acres  and  decimals  of 
an  acre  by  the  rules  for  multiplication  and  division  of  decimals. 
Thus: 

(1).   11.25  X  2.50  =  28.1250  sqch.; 
28.1250 -=-10  =  2.81250  acres. 

(2).     21.32 X 8=  170.56  sq.ch.; 
170.56  -i- 10  =  17.056  acres. 

(3).     22X15  =  330  sq.ch.; 
330  -=-10  =  33.0  acres. 

( 193).  The  following  special  rules  are  deduced  from  the  pre- 
ceding processes : 

(1).  When  each  dimension  contains  hundredths  of  a  chain 
(links),  the  product  may  be  reduced  to  acres  by  pointing  off  five 
decimal  places. 

(2).  When  one  contains  hundredths  and  the  other  is  expressed 
in  full  chains,  the  product  is  reduced  to  acres  by  pointing  off 
three  decimal  places. 

(3).  When  each  dimension  is  in  full  chains,  reduce  to  acres  by 
pointing  off  one  decimal  place. 

If  either  dimension  contain  a  fraction  of  a  link,  point  off  as 
many  additional  places  in  the  prodnct  as  are  necessary  to  express  the  frac- 
tion. Thus : 

(1 ).   5.125  X  3.20  =  1 .640000  acres. 

(2).   2.3725X5  =  1.18625  acres. 
(105) 


106  MANUAL  OF  PLANE  SURVEYING. 

EXAMPLES. 

(194).        (1).  21.52     X  12.40    =  how  many  acres? 
(2).   40.24     X  20.31    =  " 
(3).   16.42     X18        =  " 
(4).    12          X13        =  "        "        " 
(5).   12.165    X 15-30  =  "        "        " 
(6).   15.2425X12.125=  " 

(  195).  Every  tract  of  land  is  in  figure  a  polygon,  and  its  area 
is  computed  according  to  the  rule  for  finding  the  area  of  the  par- 
ticular polygon  representing  its  contour. 

( 196 ).  EECT  ANGLES. 

1.  Multiply  the  length  by  the  breadth  and  the  product  will  be 
the  area. 

2.  If  the  rectangle  be  a  square,  the  area  may  be  found  by  squar- 
ing one  side. 

( 197 ).  PARALLELOGRAMS. 

Multiply  the  length  of  one  side  by  the  perpendicular  distance 
between  this  side  and  the  opposite.  The  product  will  equal  the 
area. 

In  Fig.  36  the  line  A  B  represents  the  perpendicular,  and  the 

AC 


£          /5.60 

FIG.  36. 

area  of  the  parallelogram  is  therefore  equal  to  the  product  of  15.60 
by  7.55=  11.778  acres. 

If  either  of  the  sides  B  C  or  D  E  be  given  instead  of  B  E  or  C  D, 
the  perpendicular  must  be  measured  in  the  direction  of  the  length 
of  the  figure. 


MANUAL  OF  PLANE  SURVEYING.  107 

( 198 ).  TRAPEZOIDS. 

Multiply  the  sum  of  the  parallel  sides  by  the  perpendicular 
distance  between  them,  and  half  the  product  will  be  the  area. 


7*.  30 


FIG.  37. 

In  Fig.  37  the  line  A  B  represents  the  distance  between  the  par- 
allel sides,  and  the  area  Is  equal  to  (  (16.24  -\-  14.90)  X  6.40)  H-  2  = 
9.4648  acres. 

When  the  trapezoid  contains  two  right-angles,  the  line  between 
them  represents  the  perpendicular  distance  between  the  parallel 
sides. 

( 199 ).  TRIANGLES. 

In  computing  the  area  of  triangles,  two  general  classes  will  be 
considered : 

1.   Triangles  whose  base  and  altitude  are  given. 

The  area  of  triangles  of  this  class  is  found  by  multiplying  the 
base  by  the  altitude  and  taking  half  the  product. 


FIG.  38. 

Fig.  38  represents  a  triangle  whose  base  is  18.05  and  altitude 
5.90.     Its  area,  therefore,  is  (18.05  X  5.90)  -=-  2  =  5.32475  acres. 
Eight-angle  triangles  belong  to  this  class. 


108  MANUAL   OF   PLANE   SURVEYING. 

When  the  three  sides  are  given,  isosceles  triangles  may  be 
brought  under  this  class  by  the  following  rule: 

Square  one  of  the  equal  sides  and  subtract  the  square  of  half 
of  the  odd  side.  The  square  root  of  the  difference  will  equal  the 
altitude. 

The  altitude  of  an  equilateral  triangle  is  found  by  extracting 
the  square  root  of  the  square  of  one  side  minus  the  square  of 
half  of  one  side. 

EXAMPLES. 

(1 ).  The  base  is  14.75  and  the  altitude  2.90.  What  is  the  area  of 
the  triangle? 

(2).  The  base  and  perpendicular  of  a  right-angle  triangle  are  5.60 
and  7.42,  respectively.  What  is  its  area? 

(3).  In  an  isosceles  triangle  the  even  sides  are  each  9.16  in 
length  and  the  odd  side  7.45.  What  is  the  area? 

(4).  What  is  the  area  of  an  equilateral  triangle  each  of  whose 
sides  is  7.25  in  length? 

2.   Triangles  whose  altitude  is  not  given. 

The  general  rule  for  triangles  of  this  class  is  the  following : 

(a).   Take  half  the  sum  of  the  three  sides. 

(6).   Subtract  from  the  half  sum  each  side  severally. 

(c).    Multiply  the  half  sum  and  three  remainders  together. 

(d).   Extract  the  square  root  of  the  product  for  the  area. 


FIG.  39. 

Let  Fig.  39  represents  a  triangle  whose  area  is  to  be  computed. 
Then,  10.54  -f  14.60  +  8.72  = 
33.86  -4-  2  =  16.93 
16.93  —  10.54  =  6.93 
16.93  —  14.60  =    2.33 
16.93  —    8.72  =    8.21 


^716.93  X  6.39  X  2.33  X  8.21  =  area. 


MANUAL   OF   PLANE   SURVEYING. 


109 


EXAMPLES. 

1.  What  is  the  area  of  a  triangle,  the  sides  of  which  are  4.20, 
2.65,  3.71,  respectively  ? 

2.  The  sides  of  a  triangle  are  2.91,  6.90,  and  5.42,  respectively. 
What  is  its  area? 

It  is  sometimes  more  convenient  to  measure  the  altitude,  and 
'lius  place  the  triangle  under  the  first  class. 

( 200 ).  TRAPEZIUMS. 

Divide  the  trapezium  into   two  triangles.     The  sum  of   their 
areas  will  be  the  area  of  the  trapezium. 
To  do  this,  measure  a  diagonal  of  the  trapezium. 


to.  7? 

FIG.  40. 
Fig.  40  represents  a  trapezium,  one  of  whose  diagonals  has  been 

A 


110  MANUAL  OF  PLANE  SURVEYING. 

measured.  It  will  be  seen  that  its  area  will  equal  the  sum  of  the* 
areas  of  the  triangles  A  B  D  and  B  C  D. 

A  serious  mistake  is  sometimes  made  by  incompetent  persons  by 
multiplying  together  the  half-sums  of  the  opposite  sides  for  the 
area. 

When  one  angle  of  the  trapezium  is  re-entrant,  as  in  Fig.  41,  the 
area  may  be  found  by  subtracting  the  area  of  the  triangle  BCD 
from  that  of  the  triangle  A  B  D ;  or  it  may  be  computed  the  same 
as  when  the  angles  are  all  salient  by  omitting  the  triangle  BCD 
and  measuring  a  diagonal  from  A  to  C. 

( 201 ).  ANY  FIGURE. 

Divide  the  figure  into  triangles  and  compute  their  areas  sepa- 
rately. The  sum  of  the  areas  of  the  triangles  will  be  the  area  of 
the  figure. 

The  area  of  the  tract  of  land  represented  in  Fig.  42  is  equal  to- 


FIG.  42. 

the  sum  of  the  areas  of  the  triangles  A  B  F,  B  E  F,  B  C  E,  and 
CDE. 

Sometimes,  when  a  tract  of  land  is  narrow  and  has  one  irregn- 
lar  boundary,  its  area  may  be  approximated  by  dividing  it  into 
trapezoids.  Fig.  43  represents  a  tract  of  this  kind  bounded  on 
one  side  by  a  creek.  In  cases  of  this  kind  the  area  of  the  tract  m 
equal  to  the  sum  of  the  areas  of  the  trapezoids  that  compose  it 


MANUAL   OF  PLANE  SURVEYING. 


Ill 


FIG.  43. 

(202). 

COMPUTATION  OF  AREA  BY  LATITUDES  AND  DEPARTURES. 

The  method  of  latitudes  and  departures  now  to  be  developed  is 
simple,  precise,  expeditious,  and  universal  in  application,  if  the 
course  and  distance  of  each  of  the  boundary  lines  of  the  tract 
whose  area  is  to  be  computed  are  given. 

(203).  In  plane  surveying,  meridians,  like  parallels  of  lati- 
tude, are  supposed  to  be  parallel  to  one  another,  and  the  latitude 
of  a  course  is  the  distance  between  two  parallels  running  through 
its  extremities,  while  the  departure  of  a  course  is  the  distance  be- 
tween two  meridians  drawn  through  its  extremities.  In  Fig.  44 
the  latitude  of  the  course  A  B  is  represented  by  B  C,  and  its  de- 


rl_m 


FIG.  44. 


112 


MANUAL   OF  PLANE  SURVEYING. 


parture  by  A  C.  It  is  evident  that  the  latitude  of  a  course  is  equal 
to  the  difference  of  latitude  of  its  extremities,  and  that  its  depar- 
ture is  equal  to  the  difference  .of  longitude  of  its  extremities. 

(  204).  The  latitudes  of  courses  bearing  north  are  called  north 
latitudes  or  northings,  and  of  those  bearing  south,  south  latitudes  or 
southings.  Likewise  the  departures  of  courses  bearing  east  are 
called  east  departures  or  eastings,  aud  of  those  bearing  west,  west  de- 
f  Tortures  or  westings. 

In  Fig.  45  the  latitudes  of  A  B  and  A  F  are  northings,  and  of 


FIG.  45. 

A  C  and  A  D  southings ;  while  the  departures  of  A  B  and  A  C  are 
eastings,  and  of  A  D  and  A  F  westings. 

(205).  North  latitudes  are  additive  and  are  marked  with  the 
sign  -)-,  plus,  while  south  latitudes  are  subtractive  and  marked  with 
the  sign  — ,  minus.  In  the  same  manner,  east  departures  are 
additive  and  marked  +,  and  west  departures  are  subtractive  and 
marked  — . 

(  206  ).  If  we  now  refer  to  Fig.  12,  we  shall  see  that  the  radius 
A  F,  may  represent  the  course  A  C,  Fig.  46,  whose  latitude  and 
departure  we  wish  to  find.  Then  will  C  E,  the  departure,  equal 


MANUAL    OF    PLANE   SURVEYING. 


113 


the  sine  of  the  angle  BAG,  and  E  A,  the  latitude,  equal  the  cosine 
of  the  angle  B  A  C. 

(  207 ).  A  Table  of  Natural*  Sines  and  Cosines  is  given  in  the 


FIG.  46. 


APPENDIX,  by  which  the  latitude  and  departure  of  any  course  may 
be  easily  found.  In  this  table  the  length  of  the  sine  and  cosine  is 
given  for  a  radius  equal  to  unity,  for  each  degree  and  minute  of 
arc  between  0  and  90°;  and,  hence,  to  find  the  latitude  of  any 
course,  it  is  necessary  only  to  multiply  the  cosine  of  its  bearing  by  the 
length  of  the  course,  and  to  find  the  departure  of  any  course,  to  mul- 
tiply the  sine  of  its  bearing  by  the  length  of  the  course.!' 

For  instance,  suppose  it  is  required  to  find  the  latitude  and  de- 
parture of  a  course  bearing  N  42°  33'  E,  and  20.22  in  length. 

By  referring  to  the  Table,  we  find  the  cosine  of  the  bearing  to 
to  be  .73669,  and  the  sine  of  the  bearing  to  be  .67623. 

Therefore,  the  latitude  of  the  course  will  equal 

.73669  X  20.22  =  14.8958, 
and  the  departure  of  the  course  will  equal 

.67623  X  20.22  =  13.6733. 

In  using  the  Table,  when  the  bearing  is  45°  or  less,  take  the  de- 
grees from  the  top  of  the  page  and  the  minutes  from  the  left-hand 

*  Called  natural  sines  and  cosines  to  distinguish  them  from  logarithmic  sines 
and  cosines. 

fin  this  rule  observe  that  the  angles  are  measured  to  the  right  and  left  of 
the  vertical  radii.  If,  as  in  Fig.  12,  they  were  measured  from  horizontal 
radii,  the  word  "sine"  would  be  used  for  "cosine,"  and  "cosine"  for 
"sine"  in  the  rule. 

8 


114  MANUAL   OF   PLANE  SURVEYING. 

column,  and  when  the  bearing  is  greater  than  45°,  use  the  degrees 
at  the  bottom  of  the  page  and  the  minutes  in  the  right-hand 
column. 

( 208 ).  EXAMPLES. 

The  course  and  distance  are  given  in  each  of  the  following  cases. 
Find  the  latitudes  and  departures  : 

(1).  N  52°  16'  W,  10.12. 

This  bearing  is  greater  than  45°;  so  the  degrees  must  be  taken 
from  the  bottom  of  the  page.  Having  found  the  double  column 
marked  52°,  ascend  it  to  the  line  marked  16'  on  the  right.  We 
now  find  the  cosine  to  be  .61199,  and  the  sine  to  be  .79087;  there- 
fore the  Latitude  =.61199  X  10.12,  and  the 
Departure  =  .79087  X  10.12. 

(2).   S  15°  40/ E,  11.41. 

(3).   N  21°  32'  W,  19.71. 

(4).  S  88°  56'  E,  73.98. 

(5).   N  66°  25' E,  46.12. 

(  209  ).  It  will  be  seen  that  the  columns  in  the  table  marked 
"sine"  at  the  top  are  marked  "cosine"  at  the  bottom,  and  that 
those  marked  "cosine"  at  the  head  are  marked  "sine"  below. 
Care  must  be  taken  to  use  the  heading  for  bearings  read  from  the 
top,  i.  e.,  for  bearings  not  greater  than  45°;  and  the  bottom  mark- 
ings for  bearings  read  from  below,  i.  e.,  for  bearings  greater  than 
45°. 

(  210 ).  TRAVERSE  TABLES  are  sometimes  used  instead  of  the 
Table  of  Natural  Sines  and  Cosines  in  determining  the  latitudes 
and  departures  of  courses,  and  somewhat  facilitate  calculations  in 
many  cases;  but  they  are  usually  computed  only  to  quarter- 
degrees,  and  it  has  been  thought  best  to  use  in  the  present  work 
only  the  more  accurate  method  of  natural  sines  and  cosines. 

(211).  In  the  survey  of  every  tract  of  land,  the  sum  of  the 
north  latitudes  should  equal  the  sum  of  the  south  latitudes,  and 
the  sum  of  the  east  departures  should  equal  the  sum  of  the  west 
departures;  and,  hence,  in  plotting  a  survey,  or  making  prepara- 
tions to  compute  the  area  of  the  tract,  we  have  an  almost  infalli- 
ble means  of  testing  the  accuracy  of  the  survey  by  which  the 
course  and  distance  of  each  of  its  boundaries  were  determined. 

(212).    Let  us  now  make  an  application  to  the  survey  of  the 


MANUAL   OF   PLANE  SUBVEYISG. 


115 


following  described  tract  of  land  :    Running  N  10°  E,  5.60 ;  thence 
S  35°  3V  E,  4.00;  thence  S  55°  30'  W,  4.00,  to  the  place  of  begin- 
ning- 
Taking  each  course  separately,  we  find  the  respective  latitudes 
and  departures. 

(1).   Latitude  of  first  course  equals  .98481  X  5.60  =  5.51. 

Departure  equals  .17366  X  5.60  =  .97. 
(2).   Latitude  of  second  course  equals  .81412  X  4.00  =  3.25. 

Departure  equals  .58070  X  4.00  =  2.32. 
(3).   Latitude  of  third  course  equals  .56641  X  4.00  =  2.26. 

Departure  equals  .82413  X  4.00  =  3.29. 

The  latitude  of  the  first  course  is  a  north  latitude,  and  must  be 
marked  +,  and  the  latitudes  of  the  second  and  third  courses  are 
south  latitudes,  and  take  the  sign  — .  Likewise,  the  departures 
of  the  first  and  second  courses  are  east  departures  and  should  be 
marked  +,  while  the  departure  of  the  third  course  is  a  west  de- 
parture and  should  be  marked  — . 

The  separate  courses,  with  the  latitude  and  departure  for  each 
one,  may  be  entered  in  a  diagram  similar  to  the  one  used  in  keep- 
ing field-notes  (Art.  160),  and  a  space  left  at  the  bottom  for  the 
footings,  as  follows : 


Sta. 

Dis 

Lat. 

Dep. 

+ 

— 

+ 

—     . 

A 

N  10°  OP  E. 

5.60 

5.51 

.97 

B 

835°  3^  E. 

4.00 

3.25 

2.32 

c 

C  55°  3ff  W 

4.00 

2.26 

329 

5.51 

5.51 

3.29 

3.29 

FIG.  47. 

(  213  ).  The  reason  why  the  east  departures  should  balance  the 
west  departures,  and  the  north  balance  the  south  latitudes  will  be 
seen  by  noticing  Fig.  48,  which  represents  the  above  tract  of  land. 
The  north  and  south  lines  represent  the  latitudes,  and  the  east 
and  west  lines  the  departures. 

(  214).  When  the  +  latitudes  balance  the  —  latitudes,  and  the 
-f-  departures  balance  the  —  departures,  as  in  the  case  just  con- 


116 


MANUAL   OF   PLAXE   SURVEYING. 


sidered,  the  survey  is  said  to  "  close."  Usually,  however,  owing 
to  slight  inaccuracies  in  sighting  the  flag,  reading  the  bearing  of 
the  line,  measuring  the  line,  or,  perhaps,  all  combined,  neither  the 
latitudes  nor  departures  balance.  If  the  disagreement  is  consider- 
able, a  re-survey  should  be  made,  as  there  is  probably  an  error 


FIG.  48. 


somewhere  in  the  work;  but  if  it  is  only  slight,  as,  for  instance,  1 
or  2  links  in  7  or  8  chains,  it  is  probably  due  to  some  unavoidable 
inaccuracy  in  the  survey,  and  may  be  corrected  by  the  following 
rule: 

Find  the  amount  of  the  error  for  each  chain,  and  distribute  it  among 
the  latitudes  or  departures,  as  the  case  may  be,  in  proportion  to  their  re- 
spective lengths.  Adding  to  those  that  are  too  small,  and  subtracting  from 
those  that  are  too  large. 


MANUAL  OF   PLANE  SURVEYING.  117 

This  will  cause  them  to  balance  and  answer  all  ordinary  pur- 


( 215 ).  The  longitude  or  meridian  distance  of  a  line  is  its  mean 
distance  from  an  initial  line  or  meridian.  Preparatory  to  finding 
the  area  of  a  tract  of  land,  this  meridian  is  conceived  to  be  drawn 
through  its  extreme  western  or  eastern  corner — usually  the  west- 
ern— and  the  longitude  of  each  of  the  courses  of  the  tract  is  com- 
puted from  this  meridian  as  a  base. 

In  Fig.  49  this  meridian  is  drawn  through  the  western  corner 


FIG.  49. 

of  the  tract,  and  the  lines,  A  B,  C  D,  E  F,  G  H,  and  M  O,  repre- 
sent the  longitudes  of  the  various  courses. 

(216).  It  will  be  observed  that  there  is  a  difference  between 
longitudes  and  departures:  The  former  show  the  mean  distance 
of  the  line  from  the  meridian,  while  the  latter  indicate  the  differ- 
ence in  longitude  of  the  two  ends  of  the  line. 

(  217  )-  By  referring  to  the  figure,  it  will  be  seen  that  the  lon- 
gitude. A  .6,  of  the  first  course  is  equal  to  half  of  its  departure, 


118  MANUAL   OF   PLANE   SURVEYING. 

a  b;  and  also  that  the  longitude,  C  D,  of  the  second  course  is  equal 
to  c  d,  which  equals  the  longitude  of  the  first  course,  plus  half  the 
departure  of  the  first  course,  plus  half  the  departure  of  the  second 
course,  and  it  may  easily  be  shown  that  the  longitude  of  any  course 
is  equal  to  the  longitude  of  the  preceding  course,  plus  half  the  departure  of 
the  preceding  course,  plus  half  the  departure  of  the  course  itself. 

( 218 ).  It  must  be  borne  in  mind  that  the  algebraic  sum  is 
meant,  and  that  west  departures,  having  the  minus  sign,  are  really 
subtractive. 

(219).  In  order  to  simplify  the  rule,  and  at  the  same  time 
avoid  fractions,  it  will  be  preferable  to  double  each  of  the  pre- 
ceding expressions  and  use  double  longitudes.  The  following  will 
then  be  the  general  rule  for  finding  the  double  longitudes  of  courses. 

The  double  longitude  of  the  first  course  is  equal  to  its  departure. 

The  double  longitude  of  the  second  course  is  equal  to  the  double  longi- 
tude of  the  first  course  -\-  the  departure  of  the  first  course  +  the  departure 
of  the  second  course. 

The  double  longitude  of  any  course  is  equal  to  the  double  longitude  of 
the  preceding  course  +  the  departure  of  the  preceding  course  +  the  de- 
parture of  the  course  itself. 

COMPUTATION  OF  AREA. 

(220).  We  are  now  prepared  to  compute  areas  by  means  of 
longitudes.  Take  for  example  the  tract  of  land  described  in 
Art.  212. 

The  area  of  the  triangle  ABC,  Fig.  50,  is  equal  to  the  area  of 
the  trapezoid  E  A  B  D,plus  the  area  of  the  triangle  BCD,  minus 
the  area  of  the  triangle  ACE. 

Finding  the  area  of  each  of  these  figures,  respectively,  we  have : 

Area  of  trapezoid  EABD  =  DEXab  =  the  product  of  the 
latitude  of  the  course  A  B  by  its  longitude  =  (3.25  X  2.13)  =  .692 
acre.  (See  Fig.  48.) 

Area  of  triangle  BCD  =  CDXef  =  the  product  of  the  lati- 
tude of  the  course  B  C  by  its  longitude  =  (2.26  X  1-645)  =  .371 
acre. 

Area  of  triangle  ACE  =  CEXcd  =  the  product  of  the  lati- 
tude of  A  C  by  its  longitude  =  (5.51  X  -485)  =  .267  acre. 


MANUAL   OF  PLANE  SURVEYING. 


119 


Therefore,  the  area  of  the  triangle  A  B  C  =  .692+  .371— .267  = 
.796  acre. 


FIG.  50. 

(  221 ).  In  computations  of  this  kind  the  product  of  a  longitude 
by  a  north  latitude  is  called  a  north  product,  and  by  a  south  latitude,  is 
called  a  south  product,  and  the  difference  between  the  north  products  and 
south  products  is  the  area  of  the  tract. 

(  222  ).  Hereafter  double  longitudes  will  be  used,  and  the  differ- 
ence between  the  north  products  and  the  south  products  will  then 
be  double  the  area  of  the  tract. 

(  223 ).  The  differeht  steps  in  the  process  of  computation  may 
be  shown  very  nicely,  and  the  work  kept  in  compact  form,  by  rul- 
ing a  sheet  of  paper  in  fourteen  columns,  adding  seven  to  the  right 
of  the  seven  shown  in  Fig.  47.  In  the  first  four  of  the  added 
seven  write  the  corrected  latitudes  and  departures,  in  the  fifth 
the  double  longitudes,  and  in  the  sixth  and  seventh  the  north 
and  south  products  or  areas  marked  -j-  and  — ,  same  as  latitudes. 


120 


MANUAL   OF   PLANE   SURVEYING. 


The  following  will  serve  as  an  illustration,  and  at  the  same  time 
indicate  the  process  used  in  computation. 


i 


+ 


I' 


S    8 


s  s 


S  IS 


e     g  |S 
o    •*    t>: 


FIG.  51. 


In  this  example  the  error  in  latitudes  and  departures  amounts 
to  very  little  in  each  case,  and  might  have  been  disregarded  in 
the  calculation. 


MANUAL   OF   PLANE   SURVEYING.  121 

( 224 ).  The  following  is  the  general  rule  for  computing  areas 
by  double  longitudes : 

Multiply  the  double  longitude  of  each  course  by  its  latitude. 
If  the  latitude  is  north  or  plus,  write  the  product  in  the  column  of  plus 
areas.     If  south  or  minus,  u'rite  the  product  in  the  column  of  minus  areas. 
Half  the  difference  between  the  sums  of  the  areas  of  these  two  columns 
will  be  the  area  of  the  tract. 

This  rule  holds  good  for  any  tract  of  land  bounded  by  straight 
lines. 

(  225  ).   When  the  most  westerly  corner  of  the  tract  can  not  be 
determined  readily  it  is  best  to  draw  a  plot  of  the  tract  according 
to  the  directions  given  in  the  chapter  on  PLOTTING. 
( 226 ).   Compute  the  areas  of  the  following  tracts : 
(1).   N34°15'E,     2.73. 
N  85°  00'  E,     1.28. 
S  56°45/E,     2.20. 
S  34°  15'  W,    3.53. 
N  56°  3(K  W,    3.20. 
(2).   S  73°15'E,   19.08. 
S  19°  30'  W,  13.68. 
N  69°  15'  W,  10.34. 
S  20°  15'  W,  11.36. 
N68°00/W,    9.06. 
N  20°  15'  E,  23.56. 

(3).  South,  3.75;  S  35°  OO'  E,  1.04;  S  86°  30'  E,  5.02;  N  82° 
00'  E,  1.72 ;  S  34°  30'  E  2.46 ;  S  77°  30'  E,  4.  25 ;  N  45° 
30'  E,  9.78;  N  2°  40X  W,  233;  West,  2.18;  N  4°  00'  E, 
1.30 ;  N  83°  45'  W,  5.35 ;  S  76°  (KK  W,  1.94 ;  S  60°  15' 
W,  2.27 ;  S  76°  00'  W,  3.47 ;  N  73°  30'  W,  2.90 ;  N  57° 
30'  W,  1.65 ;  S  21°  00'  W.  2.67. 

(  227  ).  It  is,  of  course,  plain  that  a  due  north  or  south  course 
has  no  departure,  and  that  its  latitude  is  equal  to  its  length. 
Likewise,  that  a  due  east  or  west  course  has  no  latitude,  and  its 
departure  is  equal  to  its  length. 

(  228 ).  It  is  not  absolutely  necessary  that  the  meridian  should 
be  drawn  through  the  most  westerly  station  in  calculating  the  con- 
tents, but  it  is  generally  more  convenient  to  compute  from  a  me- 
ridian so  drawn.  Sometimes  the  surveyor  imagines  the  meridian 
to  pass  through  the  most  easterly  station. 


122  MANUAL   OF   PLANE   SURVEYING. 

If  necessary,  the  areas  may  be  expressed  in  the  ordinary  de- 
nominations of  land  measure  (acres,  roods,  and  rods),  instead  of 
in  acres  and  decimals  of  an  acre,  by  reducing  the  decimals  to  in- 
tegers. Thus,  .82  of  an  acre  =  (.82  X  4)  =  3.28  R.  .28  X  40  = 
11.20  sq.  rods.  Hence,  .82  acre  =  3  K.  11.2  rods. 

QUESTIONS  ON  CHAPTER  XII. 

1.  Why  should  the  length  of  lines  be  written  in  chains  and 

links? 

2.  When  links  are  multiplied  by  links,  how  many  decimals 

are  pointed  off  in  reducing  the  product  to  acres?     Links 
by  chains  ?     Chains  by  chains  ? 

3.  Give  the  rule  for  finding  the  area  of  a  rectangle.     A  paral- 

lelogram.    A  trapezoid. 

4.  How  is  the  area  of  a  triangle  found  when  the  base  and  al- 

titude are  given?     When  the  three  sides  are  given? 

5.  How  may  the  altitude  of  an  isosceles  triangle  be  found  ?  Of 

an  equilateral  triangle? 

6.  State  the  rule  for  finding  the  area  of  a  trapezium. 

7.  When  may  the  area  of  a  figure  be  computed  by  dividing  it 

into  trapezoids? 

8.  What  is  meant  by  the  latitude  of  a  course?     Departure? 

9.  WThat   are   north   latitudes?     South    latitudes?     East   de- 

partures ?     West  departures  ? 

10.  Describe  the  Table  of  Natural  Sines  and  Cosines. 

11.  How  do  you  find  the  latitude  of  a  course  from  the  Table? 

The  departure? 

12.  Give  the  rule  for  correcting  latitudes  and  departures. 

13.  What  is  meant  by  the  longitude  of  a  line?     What  is  the 

difference  between  the  longitude  of  a  line  and  its  depar- 
ture? 

14.  State  the  rule  for  finding  the  double  longitudes  of  courses. 

15.  WThat  are  north   products   or   areas?     South   products  or 

areas  ? 

16.  Give  the  general  rule  for  computing  areas  by  double  longi- 

tudes. 


CHAPTER  XIII. 

LAYING  OUT  AND  DIVIDING  UP  LAND. 

(  229  ).  No  general  rule  can  be  given  either  for  laying  out  or 
dividing  up  land,  and  in  the  present  chapter  only  the  most  com- 
mon cases  that  arise  in  practice  will  be  considered.  This  is  nec- 
essary in  order  to  keep  our  work  within  its  intended  limits  as  well 
as  to  avoid  the  confusion  that  a  multiplication  of  details  would 
cause. 

As  a  general  thing,  a  little  ingenuity  on  the  part  of  the  surveyor 
will  enable  him  to  devise  a  method  to  meet  the  exigencies  of  the 
case  he  may  have  on  hand  when  it  can  not  be  reached  by  any  of 
the  rules  given  in  this  chapter. 

(  230 ).  In  the  problems  now  to  be  taken  up,  the  area  and  one 
or  more  of  the  boundaries  are  in  nearly  all  cases  supposed  to  be 
known,  and  it  is  required  to  find  from  these  the  length  of  certain 
other  boundary  lines  necessary  to  a  survey  of  the  tract.  The  pro- 
cesses used  in  work  of  this  kind  are  generally  the  reverse  of  those 
employed  in  the  last  chapter,  so  that  a  careful  study  of  operations 
in  computing  areas  will  materially  assist  in  the  work  now  before  us. 

LAYING  OUT  LAND. 

(  231 ).  To  Lay  Out  a  Square.— The  square  root  of  the  area  ex- 
pressed in  square  chains  and  decimals  will  represent  the  length 
of  one  of  its  sides. 

Thus,  each  side  of  a  square  tract  of  land  containing  5  acres 
equals  v/50  =  7.07. 

EXAMPLES. 

1.  What  is  the  length  of  each  side  of  a  square  tract  containing 
1  acre? 

2.  A  piece  of  land  in  the  form  of  a  square  contains  11  A.  3  R. 
26  P.     What  is  the  length  of  its  sides? 

(123) 


124  MANUAL   OF   PLANE   SURVEYING. 

(  232  ).  To  Lay  Out  a  Rectangle.— Divide  the  area  by  the  length 
of  the  given  side.  The  quotient  will  be  the  length  of  the  required 
side. 

Thus,  if  a  rectangle  contain  4  acres,  and  the  given  length  be  5 
chains,  the  length  of  the  required  side  will  equal  (40  -*-  5)  =  8 
chains. 

EXAMPLES. 

1.  The  area  of  a  rectangle  is  14  acres,  and  the  length  15.00, 
what  is  the  breadth  ? 

2.  The  area  of  a  rectangular  tract  of  land  equals  10  A.  2  R  20 
P.,  and  it  is  63  rods  long.     What  is  the  breadth  ? 

Process— 10  A.  2  E.  20  P.  =  10.625  acres. 

63  rods  =  15.75. 
(10.625  X  10)  -s-  15.75  =  6.746. 

(  233 \.  To  Lay  Owl  a  Parallelogram.— (a).  Divide  the  area  by 
the  given  length.  The  quotient  will  be  the  perpendicular  distance 
between  the  given  sides.  (Art.  197). 

(6).  Find  one  of  the  angles  of  the  parallelogram  according  to 
the  methods  explained  in  Articles  (22)  and  (23). 

( c ).  Divide  the  length  of  the  perpendicular  by  the  sine  of  the 
angle  thus  found,  and  the  quotient  will  equal  the  required  side. 

If  the  angle  of  the  parallelogram  be  greater  than  90°,  its  sup- 
plement* must  be  used  in  its  stead. 

Let  Fig.  52  represent  the  parallelogram  to  be  laid  out ;  its  area 


BACKS 


A    C  12  .  o  o 

FIG.  52. 

being  6  acres,  and  the  length  of  the  side  A  B,  12.00.  Dividing 
(6  X  10)  =  60  sq.  chains  by  12,  we  find  the  perpendicular  C  D,  to 
be  5.00  long. 

*  The  sine  of  an  angle  is  always  equal  to  the  sine  of  the  supplement  of  the 
angle. 


MANUAL   OF   PLANE   SURVEYING. 


125 


Suppose  the  angle  A  D  E  to  equal  108°;  then  its  supplement 
will  be  (180°—  108°)  =  72°,  the  sine  of  which  is  .95106. 

Dividing  5.00  by  .95106,  we  find  the  length  of  the  required  side, 
A  D,  to  be  5.257. 

Had  the  angle  BAD  been  used,  instead  of  A  D  E,  the  supple- 
ment need  not  have  been  taken,  as  it  is  less  than  90°. 

EXAMPLES. 

1.  In  a  parallelogram,  the  area  is  12  acres,  the  length  of  one 
side  14.00,  and  the  measured  angle  equal  to  61°.    What  is  the 
length  of  the  required  side? 

2.  A  field  in  the  form  of  a  parallelogram  is  22.00  long  and  con- 
tains 25  acres.     The  size  of  one  of  the  angles  is  96°  45' ;  what  is 
the  length  of  the  required  side? 

(  234 ).    To  Lay  Out  a  Eight-Angled  Triangle.— Let  it  be  required 


0 


S 


FIG.  53. 


to  lay  out  a  right-angled  triangle  containing  .3  acre  by  a  line  per- 
pendicular to  A  B,  Fig.  53. 

(a).    Measure  the  angle  BAG  and  find  its  sine. 

(6).  Multiply  the  sine  by  any  length  of  base,  as  A  D,  less  than 
the  required  base.  The  product  will  represent  the  altitude  D  E 
of  the  triangle  A  D  E. 

(c).  Compute  the  area  of  this  triangle,  and  the  length  of  the 
side  A  B  may  be  found  by  the  following  proportion : 

Area  A  D  E  :  Area  A  B  C  :  :  ( A  D)2 :  (A  B)J. 

If  A  D  =  1.20,  and  D  E  .80,  the  area  of  the  triangle  A  D  E  will 
be  .048  acre.  Then, 

.048 :  .3  : :  (1.20)2 :  (A  B)2 ;  whence  (AB)2  =  9,  and  A  B  =  3  chains. 


126 


MANUAL   OF   PLANE  SURVEYING. 


The  length  of  the  side  A  C  may  be  found  by  a  similar  propor- 
tion: 

Area  A  V  E :  Area  A  B  C : :  (A  E)2 :  (A  C)2. 

(  235 ).  To  Lay  Out  a  Trnpezoid.— Approximate  the  distance  be- 
tween the  parallel  sides  by  treating  it  as  a  parallelogram.  The 
distance  thus  found  will  be  too  short  if  the  sides  not  parallel  con- 
verge, and  too  long  if  they  diverge.  Let  Fig.  54  represent  a  tract 
to  be  laid  out  or  parted  off. 


B 


FIG.  54. 


By  dividing  the  area  by  the  length  of  the  line  A  B,  the  perpen- 
dicular is  found  to  equal  C  D.  The  guess  line  E  D  is  then  meas- 
ured, and  the  area  of  the  trapezoid  A  B  D  E  computed.  The  de- 
ficiency of  area  is  then  added  outside  of  D  E,  and  the  trapezoid 
A  B  F  G  will  then  contain  the  required  amount  of  land.  If  it  still 
vary  a  little  from  the  exact  amount,  the  line  F  G  may  be  moved 
further  out  or  in,  as  the  case  may  be.  • 

In  case  of  divergence  of  the  sides,  the  overplus  of  area  must  be- 
subtracted  from  the  computed  area,  and  the  guess  line  moved 
back  instead  of  further  out. 


FIG.  55. 


MANUAL   OF   PLANE   SURVEYING.  127 

When  the  difference  in  length  of  the  parallel  sides  can  be  deter- 
mined without  a  measurement,  no  guess  line  need  be  surveyed, 
providing  the  distance  between  the  parallel  sides  be  known.  Let 
A  B  C  D,  Fig.  55,  represent  a  tract  of  land  to  be  laid  out  or  parted 
off  the  main  tract  by  a  line  perpendicular  to  its  parallel  sides. 

Divide  its  area  by  the  perpendicular  distance  between  its  par- 
allel sides.  The  quotient  will  be  the  mean  length  of  the  trapezoid. 
From  this  subtract  half  the  distance  D  E  for  the  shorter  side,  and 
add  for  the  longer  side. 

(236).  To  Lay  Out  any  Figure.— 'When  the  underlying  princi- 
ples of  the  particular  problem  differ  from  those  obtaining  in  any 
of  the  cases  considered,  it  will  probably  be  best  to  depend  on  cor- 
rections made  from  guess  lines,  as  in  Art.  235,  and  thus  reach  the 
result  by  approximations.  Yet,  in  many  instances,  easy  and  beau- 
tiful solutions  may  be  reached  by  close  observation  and  study. 

DIVIDING  UP  LAND. 

(237  ).  Problems  in  dividing  up  land  are  such  as  grow  out  of 
division  of  estates,  generally  among  heirs.  This  division  is  made 
with  reference  to  the  value  of  the  respective  shares  (considering 
location,  improvements,  quality  of  soil,  etc.),  and  not  with  regard 
to  the  quantity  of  land  each  share  contains.  If,  however,  taking 
all  these  things  into  consideration,  the  value  of  the  land  is  uni- 
form throughout  the  tract  to  be  divided,  and  the  shares  of  the  per- 
sons among  whom  it  is  to  be  partitioned  are  equal  to  each  other, 
each  should  receive  the  same  quantity. 

In  making  a  partition  of  land  no  share  should  be  taken  out  in 
such  a  way  that  it  will  injure  any  other  share,  when  it  possibly 
can  be  avoided. 

(238).  The  problems  in  dividing  land  introduced  into  this 
chapter  are  of  the  nature  of  those  that  usually  come  up  in  prac- 
tice where  the  land  has  been  surveyed  according  to  the  Rectangu- 
lar System.  Only  simple  ones  have  been  chosen. 

(  239  ).  To  Divide  a  Rectangle  into  Equal  Parts  by  Lines  Parallel  to 
a  Side. — Divide  each  of  the  lines  upon  which  all  of  these  parts  are 
to  rest  into  as  many  equal  sections  as  there  are  shares.  Connect 
the  extremities  of  these  sections  by  perpendiculars,  and  these  per- 
pendiculars will  be  the  division  lines  of  the  shares. 


128  MANUAL   OF   PLANK  SURVEYING. 

Let  Fig.  56  represent  the  south  half  of  a   quarter-section  con- 
ace 


10.00 

10.00 

10.00 

10.00 

20 

20 
A. 

i 

I 

20 
A. 

20 
A. 

10.00 

! 
10.00       | 

10.00 

10.00 

i                  d 

/ 

FIG. 

56. 

taining  exactly  80  acres.     The  perpendiculars,  a  b,  e  d,  and  e  f, 
divide  it  into  four  parts,  each  containing  20  acres. 

( 240 ).  To  divide  a  rectangle  into  any  number  of  unequal  parts  bear- 
ing a  given  relation  to  one  another,  by  lines  running  parallel  to  a  side. — 
Suppose  that,  on  account  of  the  varying  value  of  the  land  in  the 
tract  to  be  divided,  the  shares  are  to  be  to  one  another  as  the  num- 
bers 1,  2,  and  5.  In  this  case,  divide  the  base  lines  into  parts 
bearing  the  same  relation  to  one  another  as  the  shares,  and  con- 


5.00 

10.00            25.00 

• 

10 

20 
A. 

' 
50 
A, 

5.00 

10.00 

25.00 

FIG.  57. 

nect  the  points  of  division  by  perpendiculars,  as  shown  in  the 
division  of  the  80  acre  tract  in  the  figure. 

Sometimes  it  is  possible  to  divide  land  of  varying  quality  so 
that  each  share  shall  contain  its  portion  both  of  the  best  and 
worst.  If  we  consider  the  unequal  division  in  Fig.  57  to  have 
been  caused  by  the  difference  in  quality  of  the  land  in  various 
parts  of  the  tract,  it  might  have  been  possible  to  make  the  shares 
all  equal  by  dividing  the  tract  in  the  direction  of  its  length. 


MANUAL   OF   PLANE   SURVEYING.  129 

The  figure  of  the  shares  may  be  almost  as  variable  as  the  quan- 
tity of  land  they  contain.  The  rectangular  form  is  preferred,  but 
of  course  can  not  always  be  preserved,  even  in  the  territory  sur- 
veyed according  to  the  Rectangular  System. 

(  241 ).  Problems.— 1.  Divide  a  quarter-section  of  land  into  five 
shares  in  the  series  1,  2,  3,  4,  5,  by  lines  running  parallel  to  a  side. 

2.  The  commissioners,  in  a  certain  partition  of  a  quarter-sec- 
tion, set  off  the  widow's  dower  of  30  acres  in  the  form  of  a  square 
in   the  south-east  corner,  and   divided  the   remainder,  by  lines 
running  north  and  south,  equally  among  five  children.     What 
was  the  width  of  each  share  ? 

3.  In  a  sale  of  a  quarter-section  for  taxes,  the  lowest  bid  was 
for  fifteen  acres,  and  this  amount  was  set  off  in  the  form  of  a  square 
in  the  north-west  corner.     A  few  years  afterward  the  remainder 
of   the   quarter  was  offered  again,  and   the   lowest  bid  was  for 
twelve  acres,  which  area  was  set  off  next  to  that  first  sold.     What 
were  the  dimensions  of  each  piece  ? 

4.  Divide  the  following  described  tract  into  four  equal  shares 
by  north  and  south  lines:     South   half   north-east  quarter,  and 
east   half  north-east   fourth   north-east  quarter,  and  south  half 
north-west  fourth  north-east  quarter. 

5.  Divide  a  quarter-section  into  7  equal  shares  by  lines  running 
north  and  south,  and  write  a  description  of  each  share,  giving 
metes  and  bounds. 

6.  Divide  the  following  tract  into  5  equal  shares  by  north  and 
south  lines,  and  write  a  description  of  each  share :     North  half 
south- west  quarter,  and  south-west  fourth  north-west  quarter,  and 
west  half  north-west  fourth  south-east  quarter,  giving  metes  and 
bounds. 

7., Divide  the  north-west  quarter,  and  north  half  north-east 
quarter,  and  north  half  north-east  fourth  south-west  quarter,  by 
east  and  west  lines,  into  3  shares  that  will  be  to  each  other  as  1, 
2,  and  3,  and  write  a  description  of  each  share,  giving  metes  and 
bounds. 

In  the  above  examples  each  tract  is  supposed  to  contain  exactly 
the  prescribed  amount  of  land,  and  the  boundaries  to  run  due 
east  and  west,  or  north  and  south,  as  the  case  may  be. 


130  MANUAL   OF   PLANE  SURVEYING. 

QUESTIONS  ON  CHAPTER  XIII. 

1.  Why  will  a  study  of  methods  used  in  computation  of  area 

assist  in  laying  ou.t  and  dividing  up  land? 

2.  How  do  you  determine  a  side  of  a  square  from  the  area?  Of 

a  rectangle  ? 

3.  Give  the  rule  for  finding  the  required  side  of  a  parallelogram. 

4.  Explain  the  method  given  for  laying  out  a  right-angle  tri- 

angle. 

5.  How  do  you  lay  out  a  trapezoid  from  the  area  and 'length  of 

one  of  the  parallel  sides? 

6.  In  partitioning  land,  which  is  considered,  quantity  of  land 

or  value? 

7.  Explain  the  method  of  dividing  a  rectangle  into  equal  parts 

by  lines  running  parallel  to  a  side. 
1     8.   In  dividing  lands,  what  figure  for  the  shares  is  preferred  ? 


NOTE  ON  CHAPTER  XIII.— The  cases  in  dividing  upland  given  in  this 
chapter  do  not  apply  except  where  the  land  has  been  surveyed  according  to 
the  Rectangular  System.  In  States  east  of  the  Mississippi  river,  excepting 
Michigan,  Wisconsin,  Illinois,  Indiana,  Ohio,  Mississippi,  Alabama  and  Flor- 
ida, and  even  in  many  instances  in  these  States,  as  well  as  in  the  ones  west 
of  the  river,  the  surveyor  must  modify  the  method  as  the  case  may  require. 


CHAPTER  XIV. 

SURVEYING  TOWN  LOTS. 

(242 ).  The  dimensions  of  town  lots  are  usually  given  in  feet, 
instead  of  in  chains  and  links,  and,  as  a  general  thing,  the  lots  are 
all  of  the  same  size  and  numbered  in  regular  order  from  1  up,  as 
shown  in  Fig.  58.  The  larger  figures  indicate  the  numbers  of  the 


FIG.  58. 


blocks,  and  sometimes  the  lots  in  each  block  are  numbered  sepa- 
rately ;  as  lot  5,  block  2 ;  lot  2,  block  9,  and  so  on.     As  the  town 
grows  from  the  original  plot,  the  lots  in  each  addition  are  fre- 
(131) 


132 


MANUAL   OF   PLANE   SURVEYING. 


quently  numbered  and  referred  to  the  particular  addition  to  which 
they  belong ;  as  lot  8,  Brown's  addition ;  lot  7,  Johnson's  addi- 
tion, etc. 

(  243).  The  survey  of  the  town  is  generally  based  on  some  in- 
dependent corner  of  the  section  in  which  it  is  situated,  and  im- 
portant corners  in  various  parts  of  the  town  should  be  marked 
with  durable  monuments.  Fig.  59  shows  a  few  lots  in  a  town  lo- 
cated on  a  section  line,  and  the  distance  is  given  from  the  section 
corner  to  the  south-east  corner  of  lot  number  1.  It  will  be  seen 


s 
e 

* 

3 

i 

2 

8 

J 

3  .ZS 

co« 


FIG.  59. 


that  stones  are  placed  at  the  north-east  corner  of  lot  number  4  and 
the  south-west  corner  of  lot  number  8.  These  will  enable  the  sur- 
veyor at  any  subsequent  time  to  make  a  survey  in  the  town  with- 
out going  to  the  section  corner  to  find  a  starting  point,  thus  saving 
him  time  and  trouble. 

(  244  ).  Lots  are  usually  rectangular  in  shape  and  about  twice 
as  long  as  they  are  wide,  but  this  is  not  always  the  case.  They 
may  be  any  reasonable  shape  or  size  that  adapts  them  to  the  plan 
of  the  town.  Likewise,  streets  and  alleys  generally  cross  one 
another  at  right  angles,  though  by  no  means  always. 

(  245 ).  The  plot  of  a  town  should  always  be  accompanied  by 
full  explanations  showing, 

(  1 ).   The  size  of  each  of  the  lots. 

( 2  ).   The  width  of  each  of  the  streets  and  alleys. 

(  3  ).   The  name  of  each  of  the  additions. 

(  4  ).  Any  other  explanations  necessary  to  determine  the  bear- 
ings of  any  of  the  lines  which  would  have  to  be  run  in  a  survey 
of  the  town. 


MANUAL   OF   PLANE  SURVEYING. 


133 


(  246  ).  Suppose  that  in  Fig.  58  the  lots  are  each  100  feet  long 
and  50  feet  wide,  the  streets  50  feet  wide,  and  the  alleys  16  feet 
wide.  Since  the  lots  are  rectangular,  the  north  and  south  lines 
are  at  right  angles  to  the  east  and  west  lines.  It  is  evident  that 
after  finding  a  starting  point,  the  surveyor  need  experience  no  dif- 
ficulty in  the  survey  of  any  of  the  lots. 

If,  for  instance,  he  wishes  to  survey  lot  number  13,  and  can  find 
no  corner  except  the  one  marked  with  a  stone  at  the  south-east  cor- 
ner of  the  town,  he  may  start  at  the  center  of  this  stone,  run  west 
266  feet  to  the  west  side  of  Main  street,  and  thence  north  166  feet 
to  the  south-east  corner  of  the  lot  to  be  surveyed.  He  can  then 
survey  the  lot  without  any  trouble.  Instead  of  running  first  west 
and  then  north,  he  may  run  first  north  to  the  south-east  corner  of 
lot  number  29,  and  thence  west  to  the  corner  of  the  lot  to  be  sur- 
veyed; or  he  may  take  other  routes. 

(  247  )    Fig.  60  represents  a  portion  of  a  town  in  which  the  lots 


are  of  different  sizes  and  shapes.  All  the  lots  west  of  Main  street 
are  50  ft.  wide,  except  number  12,  which  is  60  ft  wide,  and  num- 
ber 14,  which  is  75  feet  wide. 

Main  street  bears  N  40°  W. 


134  MANTTAL  OF  PLANE  SURVEYING. 

Lots  number  15,  16,  17,  18  and  19  do  not  belong  to  the  regular 
plot  of  the  town,  and  are  called  out-lots. 

The  two  alleys  running  north  into  Main  street,  and  the  one  be- 
tween lots  4  and  5  are  each  20  ft.  wide,  and  Main  street  and  the 
short  street  between  lots  2  and  3  are  each  50  ft.  wide. 

A  stone  monument  marks  the  south-east  corner  of  lot  number  13. 

The  width  of  each  tier  of  lots  is  marked  in  feet  at  the  foot  of 
the  tier. 

( 248 ).  It  is  now  a  very  easy  matter  to  survey  any  of  the  lots 
in  the  regular  plot  of  the  town.  Take,  for  example,  number  11. 
To  survey  this  lot,  measure  first  west  280  ft.,  and  thence  north  200 
ft.  to  its  south-west  corner.  From  this  point  set  off  a  perpendicu- 
lar and  extend  it  to  the  street ;  then  measure  north  50  ft.  further 
and  set  off  another  perpendicular  as  before. 

(  249  ).  EXAMPLES. 

1.  How  would  lot  number  1  be  surveyed? 

2.  Explain  a  method  of  surveying  lot  number  14. 

3.  If  the  out-lots  east  of  Main  street  were  separated  by  lines 
perpendicular  to  the  street,  what  would  be  the  bearing  of  the 
lines? 

(  250 ).   Town  lots  are  measured  either  with  a  chain  or  tape. 

The  chain  used  for  this  purpose  is  usually  50  or  100  feet  long 
and  divided  into  links,  each  1  foot  in  length.  It  is  made  light,  and 
as  greater  accuracy  is  generally  required  in  surveying  town  lots 
than  in  ordinary  surveying,  its  length  should  be  frequently  tested 
by  comparison  with  a  standard  measure.  The  length  of  the  chain 
is  affected  by  wear,  temperature,  and  accidents. 

All  measures  used  in  surveying  should  be  subjected  to  frequent 
tests.  Even  in  surveys  where  tolerable  accuracy  is  sufficient, 
there  is  no  excuse  for  neglecting  anything  that  would  be  conducive 
to  greater  accuracy. 

Tapes  used  in  measuring  town  lots  usually  consist  of  a  jointed 
steel  ribbon,  but  sometimes  a  linen  tape  through  which  a  fine 
brass  wire  is  interwoven  with  the  thread,  is  used.  Common  linen 
tapes  contract  when  wet  and  are  not  trustworthy.  The  steel  tape 
is  the  best. 


MANUAL  OF   PLANE  SURVEYING.  135 

QUESTIONS  ON  CHAPTER  XIV. 

1.  How  are  town  lots  numbered  ? 

2.  Upon  what  is  the  survey  of  a  town  usually  based  ? 

3.  What  advantage  is  there  in  having  important  corners  marked 

by  monuments? 

4.  What  is  the  usual  shape  of  town  lots  ? 

5.  State  the  explanations  that  should  accompany  the  plot  of  a 

town. 

6.  A  street  bears  N  29°  32'  E.     What  is  the  bearing  (obverse 

and  reverse)  of  a  line  perpendicular  to  it? 

7.  What  kind  of  measures  are  employed  in  the  survey  of  town 

lots? 

8.  How  is  the  length  of  the  chain  affected  ? 


CHAPTER    XV. 

PLOTTING. 

(  251 ).  Plotting  is  the  operation  of  drawing  to  a  scale  upon 
paper  the  lines  of  a  survey,  so  that  the  plot  will  be  a  correct  rep- 
resentation of  the  actual  lines  surveyed. 

(  252  ).  The  instruments  used  in  plotting  are  a  drawing  board, 
t-square,  ruler,  drawing  pen,  dividers,  protractor,  and  a  diagonal 
scale. 

1.  The  drawing  board  should  be  made  of  pine,  and  its  surface 
should  be  perfectly  smooth  and  level.     The  paper  is  fastened  to 
the  drawing  board  while  the  plot  is  drawing.     Perhaps  the  most 
suitable   size  for  a  drawing  board   is   about   30  inches   square, 
but  24  by  28  inches  makes  a  very  nice  board.     The  paper  should 
be  stretched  evenly,  and  the  edges  pulled  down  over  the  edges  of 
the  board  and  glued  or  tacked. 

2.  The  t-square,  so  called  on  account  of  its  resemblance  to  the 
letter  T,  consists  of  a  thin  blade  with  parallel  edges,  to  which  is 
attached  a  cross-head  somewhat  thicker  than  the  blade,  so  as  to 
form  a  shoulder.     The  blade  is  usually  about  24  inches  in  length, 
and  the  cross-head  about  10  inches.     By  laying  the  blade  on  the 
paper  and  pressing  the  shoulder  against  the  edge  of  the  drawing 
board,  as  shown  in  Fig.  61,  perpendiculars  may  be  drawn  to  any 
edge  of  the  paper. 

3.  A  good  box-wood  ruler,  about  12  inches  long,  divided  to  16ths 
of  an  inch,  will  answer  every  purpose  in  plotting.     This  ruler 
should   have  one  beveled  edge,   upon   which   the   divisions   are 
marked,  and  one  projecting  edge,  along  which  the  pen  should  be 
pressed  in  drawing  lines. 

4.  The  drawing  pen  consists  of  two  steel  blades,  whose  distance 
apart  is  regulated  by  a  thumb-screw.     A  little  practice  will  en- 

(136) 


MANUAL  OF   PLANE  SURVEYING. 


137 


able  any  person  to  draw  nice  smooth  lines  of  any  desirable  width 
with  the  drawing  pen.  India  ink  should  be  used,  as  it  flows  more 
smoothly  from  the  pen  than  common  ink. 


I 


FIG.  61. 

5.  The  dividers  or  compasses  is  an  instrument  used  in  drawing 
arcs,  sub-tending  angles,  etc.,  and  consists  of  two  arms  which  open 
and  shut  by  a  hinge  joint  at  the  end.  Each  of  these  arms  termi- 
nates in  a  sharp  point,  and  one,  if  not  both,  is  usually  jointed  so 
as  to  permit  the  point  to  be  taken  out  and  a  drawing  pen  put  in 
its  stead.  In  drawing  large  arcs  a  lengthening  bar  is  inserted  in 


FIG.  62. 

the  jointed  arm.     Fig.  62  represents  a  pair  of  plain  dividers.     It 
will  be  seen  that  the  arms  are  not  jointed  in  this  pair. 

6.  The  protractor  is  an  instrument  used  in  laying  out  angles. 
It  is  usually  nothing  more  than  a  semicircle  divided  to  degrees, 
half-degrees,  or  quarter-degrees.  The  degrees  are  numbered  from 
0°  to  180°  in  one  or  both  directions  from  opposite  extremities  of 
the  arc.  The  best  protractors  are  made  of  silver  or  German  silver, 
but  the  more  common  ones  are  made  of  brass  or  horn,  and  some- 
times of  paper.  Fig.  63  represents  a  small  protractor. 


138 


MANUAL   OF   PLANE  SURVEYING. 


Where  great  accuracy  is  required,  protractors  are  supplied  with 
an  arm  to  which  a  vernier,  like  the  compass  vernier,  is  attached. 

Sometimes  rectangular  protractors  are  used  instead  of  semi- 
circular. 

7.  The  diagonal  scale  of  equal  parts  is  a  flat  scale  a  given  num- 


FIG.  63. 

ber  of  units,  say  inches,  in  length,  and  has  the  space  devoted  to 
one  unit  at  the  end  divided  by  diagonals  as  shown  in  Fig.  64. 
These  diagonals  with  the  assistance  of  the  lines  running  parallel 
to  the  edges  of  the  scale,  enable  a  person  to  take  the  length  of  a 


3,8 J.  8.5.^.  3.2.7 


FIG.  64. 

line  to  T^y  of  the  unit  of  the  scale.  If  the  unit  of  the  scale  be 
1  inch,  then  the  length  of  a  line  may  be  taken  to  TJ^  of  an  inch. 
(  253  ).  In  drawing  plots  and  maps  a  unit  of  the  scale  repre- 
sents a  certain  number  of  units  of  the  line  to  be  represented  on  the 
plot.  Suppose  the  real  line  is  20  chains  long,  and  it  is  to  be  plot- 
ted to  a  scale  of  5  chains  to  the  inch.  The  line  on  the  plot  will 
therefore  be  (20  -4-  5)  =  4  inches  long.  Fig.  65  represents  a  line 
1  chain  in  length  plotted  to  different  scales. 


MANUAL  OF   PLANE  SURVEYING.  139 

1  in.  =  5.00. 


1  in.  =  2.00. 


1  in.  =  1.00. 
FIG.  65. 

( 254 ).  Let  us  now  employ  the  drawing  instruments  in  plot- 
ting lines. 

Suppose  an  east  and  west  line  2.27  in  length  is  to  be  drawn  to 
a  scale  of  1  chain  to  an  inch. 

Arrange  the  drawing  paper  with  its  edges  parallel  to  the  edges 
of  the  board,  and  then  place  the  t-square,  as  shown  in  Fig.  61, 
with  its  shoulder  fitting  squarely  to  the  left-hand  edge,  and  the 
edge  of  the  blade  just  moved  up  to  the  point  from  which  the  line 
is  to  be  drawn.  Then  spread  the  dividers  so  that  when  one  arm 
is  placed  two  units  from  the  inner  edge  of  the  divided  unit  and 
on  the  line  marked  .07,  the  other  will  just  reach  the  point  where 
this  line  crosses  the  line  marked  .2  at  the  top  of  the  scale.  The 
arms  then  embrace  the  proper  length  of  line. 

Next  place  one  arm  of  the  dividers  against  the  t-square  with  its 
point  on  the  point  from  which  the  line  is  to  be  drawn  and  swinging 
the  free  arm  round  in  the  proper  direction  until  it  too  touches  the 
same  edge  of  the  blade.  Connect  these  two  points  by  a  line  with 
the  drawing  pen,  and  it  will  be  the  required  line. 

In  like  manner,  a  line  1.25  long  to  the  same  scale  may  be  em- 
braced by  the  dividers  by  placing  one  point  at  a  on  the  diagonal 
scale,  and  the  other  at  e;  and  a  line  1.40  long  by  placing  one 
point  at  E  and  the  other  at  the  point  marked  .4  at  the  top  of  the  scale. 

If  the  diagonal  scale  is  not  long  enough  to  permit  the  required 
line  to  be  taken  off,  it  may  be  extended  by  means  of  a  ruler. 

(255).  EXAMPLES. 

Draw  lines  representing  the  following  distances : 

(1).  2.50;  scale  — 1  in.  =    1.00. 

(2).  3.79 ;  scale  —  1  in.  =    1.00. 

(3).  4.75;  scale  —  1  in.  =    2.00. 

(4).  6.42;  scale  —  1  in.  ==    5.00. 

(5).  10.00;  scale  —  1  in.  =    5.00. 

(6).  12.31 ;  scale  —  1  in.  =  10.00. 


140  MANUAL   OF   PLANE  SURVEYING. 

(  256).  Rectangular  tracts  of  land  may  be  plotted  with  the  in- 
struments used  in  drawing  lines  already  described. 

Take,  for  instance,  a  rectangular  tract  7.15  long,  4.35  wide. 

First  draw  a  line  representing  the  length  of  the  tract,  then 
another  perpendicular  to  this  at  one  end  representing  the  breadth, 
then  from  the  end  of  this  another  parallel  to  the  first  and  of  equal 
length,  and  close  by  connecting  the  extremities  of  the  first  and 
third  with  one  another. 

If  no  diagonal  scale  is  at  hand,  a  common  ruler  will  answer  for 
rough  work. 

(  257  ).  The  protractor  is  used  in  nearly  all  cases  where  the 
courses  and  distances  of  the  boundaries  of  the  tract  to  be  plotted 
are  given,  and  its  use  will  now  be  explained. 

The  bearing  of  a  line  represents  the  angle  the  line  makes 
either  with  the  magnetic  meridian  or  the  true  meridian  drawn 
through  the  point  from  which  the  bearing  is  taken  ;  and  to  deter- 
mine this  angle  and  represent  the  line  on  the  plot,  meridians 
should  be  drawn  through  each  station  of  tne  survey  as  soon  as  the 
station  is  located. 

Begin  at  any  important  station  to  draw  the  plot  by  laying  out  a 
meridian  on  the  proper  part  of  the  paper  and  locating  the  station 
on  this  meridian;  then  draw  the  first  course  at  the  proper  angle 
with  this1  meridian,  producing  it  the  required  distance,  as  explained 
in  Art.  254;  then  draw  another  meridian  through  the  other  ex- 
tremity of  the  course,  and  lay  out  the  second  course  in  the  same 
way  ;  proceed  in  this  way  until  the  lines  are  all  drawn,  and  the  last 
line  should  terminate  at  the  station  taken  as  the  starting  point  in 
the  plot. 

(  258  ).  Let  us  now  draw  a  plot  of  the  following  tract  of  land: 
N  62°  45'  E,  9.25 ;  thence  S  36°  E,  7.60 ;  thence  S  45°  30'  W,  10.40 ; 
thence  N  31°  30'  W,  10.00. 

In  this  case  we  may  commence  at  the  first  station  in  the  de- 
scription. Draw  a  meridian  as  N  S,  Fig.  66  and  locate  the  sta- 
tion at  some  point,  as  A,  on  the  meridian.  Then  place  the  pro- 
tractor so  its  center  will  fall  on  the  station  and  its  edge  coincide 
with  the  meridian,  and  with  the  point  of  a  pin  mark  the  termina- 
tion of  an  arc  of  66°  45'  from  the  north  end  of  the  protractor. 
Then  draw  the  line  A  B  from  the  station  through  this  point  and 
determine  its  length  by  the  method  explained  in  Art.  254.  Draw 


MANUAL   OF   PLANE   SURVEYING. 


141 


B  C,  C  D,  and  D  A  in  the  same  manner,  and  the  plot  will  be  com- 
plete. 

The  plot,  Fig.  66,  is  constructed  to  a  scale  of  5  chains  to  the 
inch. 

In  northerly  courses  the  angle  or  bearing  should  be  read  from 
the  north  end  of  the  protractor,  and  in  southerly  courses  from  the 
south  end. 


N 


B 


D 


fo 


FIG.  66. 


If  the  last  course  lack  but  a  little  of  terminating  at  the  first 
station,  the  discrepancy  may  be  the  result  of  the  imperfection  of 
the  instruments  employed;  but  if  the  extremities  of  the  lines  are 
a  considerable  distance  apart,  it  is  probable  that  a  mistake  has 
been  made  somewhere.  If  it  be  tested  by  latitudes  and  departures, 
and  they  balance  (Art.  212),  the  mistake  is  in  the  plot,  but  if  they 
do  not  balance,  the  error  is  in  some  of  the  previous  work. 


142 


MANUAL  OP  PLANE  SURVEYING. 


The  descriptions  given  in  Art.  226  may  be  used  as  examples  in 
plotting. 

Various  other  methods  are  also  used  in  drawing  plots,  but  the 
one  given  is,  perhaps,  the  most  speedy  and  simple,  and  will 
answer  every  purpose. 

(259).  The  pantograph  is  an  instrument  used  for  copying 
plots,  etc.,  either  in  a  reduced  or  enlarged  form.  It  consists  of 
four  rulers  arranged  somewhat  in  the  form  of  a  parallelogram.  By 
fastening  the  instrument  on  the  drawing-board  and  moving  a 
point  on  one  arm  along  the  plot  to  be  copied,  another  arm  to 
which  a  pencil  is  attached  sketches  a  precise  copy  on  the  sheet 
placed  under  it. 

(260 ).  Buildings,  springs,  etc.,  may  be  located  on  the  plot  if 
their  courses  from  certain  points  on  the  boundaries  of  the  tract 
are  known. 

For  example,  in  surveying  the  west  half  of  a  quarter-section,  a 
line  from  the  north-east  corner  to  the  north-west  corner  of  a  house 
was  found  to  bear  S  45°  W,  and  one  from  a  point  12.00  west  of  the 
north-east  corner  was  found  to  bear  S  13°  E.  While  constructing 
the  plot  these  lines  may  be  laid  out  from  the  proper  places  with 
the  protractor,  and  the  place  where  they  meet  will  be  the  north- 
west corner  of  the  house,  as  shown  in  Fig.  67. 


FIG.  67 


MANUAL  OF  PLANE  SURVEYING.  143 

( 261 ).  Plots  and  maps  may  be  colored  with  crayon  pencils  or 
with  water-colors;  but  when  water-colors  are  used  care  must  be 
taken  to  keep  them  from  running  into  one  another  and  injuring 
the  shades.  The  paper  should  be  dampened  preparatory  to  ap- 
plying them.  For  inexperienced  persons,  crayons  will  prove  the 
most  satisfactory. 

QUESTIONS  ON  CHAPTER  XV. 

1.  What  is  plotting? 

2.  Name  the  instruments  used  in  plotting. 

3.  Descr-ibe  the  diagonal  scale. 

4.  Describe  the  method  of  using  the  diagonal  scale. 

5.  Give  the  length  of  each  of  the  following  lines  plotted  ta 

scales  of  1  chain  to  an  inch,  2  chains  to  an  inch,  and  10 
chains  to  an  inch:     12.50;  15.00;  18.375;  11.25. 

6.  Describe  the  method  of  using  the  protractor. 

7.  Why  should  the  last  course  in  a  plot  terminate  at  the  first 

station  ? 

8.  For  what  is  the  pantograph  used  ? 

9.  How  may  buildings  and  other  objects  be  located  on  a  plot? 

10.  How  are  plots  and  maps  colored? 

11.  Draw  plots  of  the  tracts  described  in  Article  (226). 


CHAPTER    XVI. 

SURVEYING  WITHOUT  A  COMPASS. 

(262).  A  great  many  surveys  can  be  made  without  the  com- 
pass, and  a  few  pages  will  now  be  devoted  to  the  consideration  of 
the  most  common  cases  in  which  it  may  be  dispensed  with.  It 
must  be  borne  in  mind,  however,  that  the  compass  could  be  ad- 
vantageously used  in  nearly  all  the  cases  here  cited,  and  that  the 
methods  given  are  intended  for  use  only  in  emergencies. 

(263).  Setting  Oorners.  —  Where  two  witness  trees  taken  to  a 
corner  can  be  found,  the  corner  may  be  located  from  them  by 
their  distances  measured  from  the  sides  upon  which  the  blazes  are 
made.  Suppose  one  tree  is  15  links  from  the  corner,  and  the  other 
19  links.  Measure  off  15  links  on  one  end  of  a  cord  and  19  links 
on  the  other  end,  and  tie  a  knot  where  the  two  measurements  ter- 
minate. Then  have  the  long  end  of  the  cord  held  against  the 


WITNESS 


blaze  on  the  more  distant  tree,  and  the  short  end  against  the  blaze 
on  the  other.  Stretch  both  ends  of  the  cord  tightly,  and  the  knot 
will  mark  the  corner,  as  shown  in  Fig.  68. 

The  corner  is  always  in  front  of  the  blazes  on  the  trees. 

The  distances  of  the  witness  trees  from  the  corner  may  be  found 
from  the  field-notes  of  the  tract. 

(144) 


OF   PLANE  SURVEYING.  145 

Where  only  one  witness  can  be  found,  the  corner  can  not  be 
located  with  certainty  without  a  compass. 

This  same  method  may  be  employed,  slightly  modified,  in  lo- 
cating a  corner  by  the  two  lines  meeting  there,  when  the  length  of 
each  line  and  the  location  of  each  of  the  corners  at  the  other  end 
of  each,  are  known. 

(  264 ).  Establishing  Lines. — When  one  corner  is  visible  from 
the  coiner  at  the  other  end  of  the  line,  a  stake  may  be  put  up,  and 
intermediate  points  on  the  line  may  be  marked  at  pleasure. 

When  one  corner  is  not  visible  from  the  other,  but  its  direction  is 
approximately  known,  the  line  may  be  "  ranged  ''  from  one  to  the 
other.  To  do  this,  put  up  a  stake  or  flag  at  the  corner  from  which 
the  line  is  ranged,  and  at  a  certain  distance,  say  50  or  100  steps, 
in  the  direction  of  the  other  corner,  set  up  another  stake  or  flag. 
Then  walk  ahead  an  equal  distance  and  set  another  stake  in  line 
with  the  first  and  second.  Proceed  in  the  same  manner,  always 
setting  the  stakes  at  equal  distances  from  one  another,  and  ranging 
the  last  one  with  the  two  previously  put  up,  until  the  other  corner 
is  reached.  The  distance  that  the  line  misses,  either  to  the  right 
or  left,  can  then  be  noted,  and  the  stakes  corrected  in  a  manner 
entirely  similar  to  that  already  explained. 

For  instance,  if  there  are  12  stakes  on  the  line,  and  it  termi- 
nates 30  links  to  the  right  of  the  corner,  each  stake  must  be  moved 
to  the  left.  The  distance  it  is  to  be  moved  is  found  by  dividing 
the  distance  missed  by  the  number  of  stakes  and  multiplying  the 
quotient  by  the  number  of  the  stake  from  the  starting  point.  In 

( 1 1  'v'  ^0  1 
this  case,  the  llth  stake  must  be  moved  to  the  left  I  — ^ —  !  =27£ 

links,  the   10th      [10*3°]  =  25  links,  and  so  on. 

The  stake  put  down  at  the  corner  at  starting  is  not  counted,  and 
the  next  is  called  the  first. 

Of  course,  the  location  of  the  corners  must  be  known  before  the 
line  can  be  established. 

(  265 ).  Setting  Out  Perpendiculars.— Almost  any  kind  of  a  con- 
trivance with  two  lines  of  sight  at  right  angles  to  one  another, 
will  answer  for  this  purpose.  It  may  be  a  sort  of  cross-staff  with 
four  upright  sights  provided  with  slits  or  threads,  two  marking 
each  line  of  sight.  The  sights  need  not  be  more  than  18  inches 
10 


146  MANUAL   OF  PLANE  SURVEYING. 

apart,  and  the  apparatus  should  be  made  to  rest  on  a  staff  about 
4£  feet  high.  It  may  be  rude  in  construction,  but  the  lines  of 
sight  should  be  exactly  at  right  angles  to  one  another. 

(  266  ).  Rectangular  tracts  of  land  may  be  readily  surveyed  in 
many  instances  with  this  instrument,  but  it  should  be  used  with 
care  in  independent  divisions  of  the  section,  as  they  are  not  often 
exactly  rectangular  in  form.  One  of  the  lines  of  sight  may  also 
be  used  in  sighting  lines,  and  is  a  good  substitute  for  the  method 
of  "  ranging  "  described  in  the  preceding  article. 

(267).  Measurements.  —  Lines  may  be  measured  with  a  cord, 
tape-line,  or  pole,  and  distances  may  be  given  in  feet  or  links,  as 
best  suit  the  case  at  hand. 


APPENDIX. 


ABSTRACT    OF    DECISIONS, 


(147) 


^    /-.-^~~ 

(rp^^^ju  tf    r-t 

s£i^^£- 

&^t_  ^ 


ABSTRACT  OF  DECISIONS 

OF    THE 

UNITED  STATES  AND  VARIOUS  STATE  COURTS 

RELATING  TO  CONTRACTS,  SURVEYS,  ETC. 


(Nearly  all  of  the  following  Decisions  have  been  taken  by  per- 
mission from  DUNN'S  LAND  DECISIONS,  a  valuable  book  for  sur- 
veyors, published  by  George  H.  Frost,  New  York.) 

BOUNDARIES. 

1.  Course  and  distance  must  yield  to  natural  and  artificial  ob- 
jects of  description.     Gaveny  m.  Hinton,  2  Greene  (Iowa)  344. 

2.  Boundaries  marked  on  the  land  are  to  govern  courses  and 
distances.     Blaisdell  vs.  Bissell,  6  Barr  (Pa.)  478. 

3.  The  lines  marked  on  the  ground  constitute  the  actual  sur- 
vey and  control  the  return  of  the  surveyor,  even  where  a  natural 
or  other  fixed  boundary  is  called  for  by  the  survey,  though  the 
space  between  the  two  is  but  twelve  perches  in  breadth.     Walker 
TO.  Smith,  2  Barr  (Pa.)  43;  Hall  m  Tanner,  4  Barr  (Pa.)  244. 

4.  A  grant  called  for  a  certain  number  of  poles  "  to  a  stake, 
crossing   the  river."     Held,  that  the   line  must  cross   the  river, 
though  the  distance  terminated  before  entering  it.     Whiteside  i». 
Singleton,  1  Meigs  (Tenn.)  207j 

5.  A  survey  must  be  closed  in  some  way  or  other.    If  this  can 
be  done  only  by  following  the  course  the  proper  distance,  then  it 
would  seem  that  distance  should  prevail ;  but  when  distance  falls 

(149) 


150  APPENDIX. 

short  of  closing,  and  the  course  will  do  it,  the  reason  for  observing 
distance  fails.     Doe  vs.  King,  3  How.  (Miss)  125. 

6.  Where  a  deed  describes  lands  by  its  admeasurements,  and, 
at  the  same  time,  by  known  and  visible  monuments,  these  latter 
shall  govern.     Mayhew  vs.  Norton,  17  Peck  (Mass.)  357;  Massen- 
gille  vs.  Boyles,  4  Humph.  (Tenn.)  205;  Woods  vs.  Kennedy,  5 
Monr.  (Ky.)  174;  Nelson  vs.  Hall,  1  McLean  (U.  S.)  518;  Camp- 
bell vs.  Clark,  8  Mis.  553. 

7.  The  rule  that  monuments  control  in  boundaries  is,  however, 
not  inflexible  ;  and  in  case  where  no  mistake  could  reasonably  be 
supposed  in  the  courses  and  distances,  the  reasons  of  the  rule  were 
held  to  fail,  and  the  rule  itself  was  not  applied.     Davis  vs.  Rains- 
ford,  17  Mass.  207. 

8.  A  line  is  to  be  extended  to  reach  a  boundary  in  the  direc- 
tion called  for,  disregarding  the  distance.     Witherspoon  vs.  Blanks, 
1  Taylor  (N.  C.)  110. 

9.  If  a  vendor  hold  two  tracts  adjoining,  and  sell  a  certain 
quantity  by  metes  and  bounds,  though  the  deeds  call  for  one  tract, 
yet  if  the  metes  and   bounds  run  into  the  other,  the  purchaser 
shall  hold  according  to  the  metes  and  bounds.     Wallace  vs.  Max- 
well, 1  J.  J.  Marsh  (Ky.)  447;  Mundell  vs.  Perry,  2  Gill  &  Johns 
(Md.)  206. 

10.  Posts  set  up  at  corners,  between  adjoining  owners  of  land, 
control  the  calls  for  course  and  distance  and  establish  the  bound- 
ary where  they  are  mentioned  and  recognized  in  the  deeds.    Al- 
shire  vs.  Hulse,  5  Ham.  (Ohio)  534. 

11.  Where  land  is  described  as  running  a  certain  distance  by 
admeasurement,  to  an  ascertained  line,  though  without  a  visible 
boundary,   such   line   will  control  the   admeasurement   and   de- 
termine the  extent  of   the  grant.     Flagg  rs.  Thurston,  13  Pick. 
(N.  Y.)  145;  Carroll  t«.  Norwood,  5  Har.  &  J.  (Md.)  163. 

12.  Where  the  line  or  course  of  an  adjoining  tract,  being  suffi- 
ciently established,  are  called  up  in  a  patent  or  deed,  the  lines 
shall  be  extended  to  them  without  regard  to  distance.    Cherry  vs. 
Slade,  3  Murph.  (N.  C.)  82. 

13.  Where  the  boundaries  of  land  are  fixed,  known,  and  un- 
questionable monuments,  although  neither  courses,  nor  distances, 
nor  the  computed  contents  correspond,  the  monuments  must  gov- 
ern.   Pernam  rs.  Wead,  6  Mass.  131 ;  Calhoun  vs.  Wall,  2  Har.  & 
McHen.  (Mo.)  416. 


APPENDIX.  151 

14.  If  a  deed  from  the  government  of  the  U.  S.,  or  an  individ- 
ual, describes  land  as  partly  bounded  by  a  river,  the  river  bound- 
ary will  be  adhered  to,  though  it  does  not  correspond  with  estab- 
lished corners  and  monuments.     Sheltonaiis.  Mauphin,  16  Mo.  124. 

15.  If  nothing  exists  to  control  the  call  for  courses  and  dis- 
tances, the  land  must  be  bounded  by  the  course  and  distances  of 
the  grant,  according  to  the  magnetic  meridian ;   but  courses  and 
distances  must  yield  to  natural  objects.     16  Ga.  141. 

17.  The  corners  established  by  the  original  surveyors  of  public 
lands  under  the  authority  of  the  United  States,  are  conclusive  as 
to  the  boundaries  of  sections  and  divisions  thereof,  and  no  error 
in  placing  them  can  be  corrected  by  any  survey  made  by  individ- 
uals or  by  a  state  surveyor.     Arnier  vs.  Wallace,  28  Miss.  556. 

18.  Whenever  natural  or  permanent  objects  are  embraced  in 
the  calls  of  either  a  survey  or  a  patent,  these  have  absolute  con- 
trol, and  both  course  and  distance  must  yield  to  them.     Brown  vs. 
Huger,  21  Howard  (U.  S.)  305. 

19.  In  determining  boundaries  under  a  grant,  natural  objects, 
.as  landmarks,  are  to  be  considered  before  courses  and  distances. 
Daggett  vs.  Wiley,  6  Fa.  482. 

20.  Where  adjoining  proprietors  abut  on  opposite  banks  of  a 
•stream,  their  boundary  line  will  follow  the  natural  and  imper- 
ceptible alterations  in  its  course,  but  not  changes  caused  by  arti- 
ficial means.     Halsey  vs.  McCormick,  3  Kernan  (N.  Y.)  296. 

21.  When  lands  are  described  in  a  deed  or  grant  as  bounded  by 
river  not  navigable,  the  center  of  the  stream  is  to  be  considered 
the  boundary.     Claremont  vs.  Carlton,  2  N.  Hamp.  369 ;  Palmer 
DS.  Mulligan,  3  Caines  (N.  Y.)  407,  319;  Hayes  vs.  Bowman,  1 
-Hand  (Va.)  417;  Ingraham  vs.  Wilkinson,  4  Pick.  (Mass.)  268; 
•Gavil  vs.  Chambers,  3  Ham.  (Ohio)  496 ;  Brown  vs.  Kennedy,  5 
Har.  &  J.  (Md.)  195;  Arnold  vs.  Mundy,  1  Halst.  (N.  J.)  1. 

QUANTITY  OF  LAND. 

22.  A  conveyance  by  metes  and  bounds  will  carry  all  the  land 
•contained  in  them.     Belden  vs.  Seymour,  8  Conn.  19;  Jackson  vs. 
Ives,  9  Cow.  (N.  Y.)  661.     Although  it  be  more  or  less  than  is 
stated  in  the  deed.     Butler  vs.  Widger,  7  Cow.  (N.  Y.)  723. 

23.  Where  a  specified  tract  of  land  is  sold  for  a  gross  sum,  the 
boundaries  of  the  tract  control  the  description  of  the  quantity  it 
contains,  and  neither  party  can  have  a  remedy  against  the  other 


for  an  excess  or  deficiency  in  the  quantity,  unless  such  excess  or 
deficiency  is  so  great  as  to  furnish  evidence  of  fraud  or  misrepre- 
sentation. Voorhees  vs.  De  Meyer,  2  Bar.  Sup.  Ct.  Rep.  (N.  Y.)  87. 
24.  Where  a  person  purchases  land  by  metes  and  bounds  said 
to  contain  a  certain  number  of  acres,  more  or  less,  he  is  entitled 
to  all  the  land  within  the  limits,  whatever  the  number  of  acres 
may  be.  Bratton  ts.  Clawson,  3  Strobh.  (S.  C.)  127. 

24.  Quantity,  although  the  least  reliable  and  last  to  be  resorted 
to  of  all  descriptions  in  a  deed,  in  determining  the  boundaries  of 
the  premises  conveyed,  may  sometimes  be  considered  in  corrobora- 
tion  of  other  proof.     McClintock  vs.  Rogers,  11  Ills.  279. 

FIGURE  OP  TRACTS  OF  LAND. 

25.  If  the  order  for  a  survey  of  land  do  not  certainly  determine 
the  form  in  which  it  should  be  made,  the  survey  ought  to  be  in  a 
square.     Kennedy  vs.  Paine,  Hardin,  10. 

26.  "Seventy  acres,  being  and  lying  in  the  south-west  corner" 
of   a  section,  is  a  good  description,   and  the  land  will  be  in  a 
square.     2  Ham.   (Ohio)   327;   Cockrell  vs.   McQuinn,   4   Monr. 
(Ky.)  63. 

The  rectangular  figure  will  be  preserved  in  preference  to 
any  other  in  fixing  locations.  Massie  is.  Watts,  6  Cranch,  148 ; 
Holmes  vs.  Trout,  7  Pet.  171. 

ACQUIESCENCE  IN  BOUNDARIES. 

28.  Acquiescence  for  a  long  time  (e.  g.  for  eighteen  years),  in 
an  erroneous  location,  is  conclusive  on  the  party  making  or  acqui- 
escing in  such  location.    Rockwell  vs.  Adams,  6  Wend.  (N.  Y.)  469. 

29.  Where  a  boundary  is  disputed  between  parties  who  own  ad- 
joining tracts,  and  the  parties  employ  a  surveyor,  who  runs  out 
the  line,  and  marks  it  on  a  plat  in  their  presence,  as  a  boundary, 
after  twenty  years  corresponding  possession,  they  are  concluded 
by  it.     Boyd  vs.  Graves,  4  Wheat.  513. 

30.  An  acquiescence  for  twenty  years  is,  as  a  general  rule,  nec- 
essary to  support  an  implied  agreement  in  respect  to  a  boundary 
different  from  that  clearly  expressed  in  the  title  deeds.     Ball  vs. 
Cox,  7  Ind.  453. 

31.  Where  two  persons  own  equal  parts  of  a  lot  of  land  in  sev- 
erally, but  not  divided  by  visible  monuments,  if  both  are  in  pos- 
session of  their  respective  parts  for  fifteen  years,  acquiescing  in  an 


APPENDIX.  153 

imaginary  line  of  division  during  that  time,  that  line  is  thereby 
established  as  a  divisional  line.  Beecher  vs.  Parmele,  9  Ver.  352 ; 
also  18  Ver.  395. 

32.  Maintaining  a  fence  for  many  years  is  strong  but  not  con- 
clusive evidence  of  limitation  of  claim  to  the  boundary.     Potts  tw. 
Everhart,  26  Pa.  493. 

33.  A  party  is  precluded  upon  principles  of  public  policy,  from 
setting  up  or  insisting  upon  a  boundary  line,  in  opposition  to  one 
which  has  been  steadily  adhered  to  upon  both  sides  for  more  than 
forty  years.     Baldwin  vs.  Brown,  16  N.  Y.  359. 

34.  A  division  fence  of  more  than  twenty-one  years'  standing, 
although  crooked,  constitutes  the  line  between  adjacent  land  own- 
ers, even  though  the  deeds  of  both  parties  call  for  a  straight  line 
between  acknowledged  landmarks.   McCoy  vs.  Hance,  28  Tenn.  149. 

35.  Where  adjoining  proprietors,  being  unable  to  ascertain  the 
division  line,  agree  verbally  upon  a  certain  line,  the  agreement  is 
binding,  and  improvements  by  one  up  to  the  line  is  notice  thereof 
to  a  purchaser  from  the  other.     Houston  vs.  Sneed,  15  Texas,  307. 

36.  An  ancient  line  of  division  marked  on  the  ground  by  ad- 
joining owners,  and  afterwards  acted  upon  by  them,  will  become 
the  boundary  between  the  lots,  although  different  from  the  line 
described  in  the  original  deeds.   Hathaway  vs.  Evans,  108  Mass.  267. 

37.  A  possession  for  twenty  years  of  a  part  of  the  land  in  dis- 
pute, in  reference  to  a  line  conflicting  with  another  tract,  of  which 
another  party  may  be  also  in  actual  possession,  but  outside  of  the 
disputed  territory,  may  be  enough  to  presume  the  execution  of  a 
deed  conveying  the  land  in  dispute  to  the  party  in  possession. 
Amick  vs.  Holman,  13  Shobh.  (S.  C.)  132. 

38.  Parties  are  not  bound  by  a  consent  to  boundaries,  which 
have  been  fixed  under  an  evident  error,  unless  perhaps  by  the  pre- 
scription of  thirty  years.     Gray  vs.  Couvillon,  12  La.  Ann.  730. 


TABLE  OF 

NATURAL  SINES  AND  COSINES. 

(See  Articles  (88),  (207),  (208),  (209),  (210),  etc.) 


(155) 


156 


TABLE   OF   NATURAL   SINKS    AND    COSINES. 


TABLE    OF    NATURAL    SIXES    AND   COSINES. 


157 


>     M.       Sine. 
<!        0     .0871 


141  33  .99011 


14119 

14148i.98994 


14117 


.!-'.<.  I 


14205  .98986 

14234 

14968 

14X98 

14890 


14378  .98961 
14-107  .98957 
14436  \989f 

14493  .98944 
14E22  .9Hi4l 
14551;. 9898 
14580  .9893: 

1  !».( >•  .!.89-,> 

MfttJ  .l>f-*.n 
14C85  .98914 
14723   !<8<<1 
14752:. 989(16 
14781  .98902 
14810  .98897 
14838   98893 
14867  .98889 
14896  .98884 
14925:. 98880 
14954!. 
14982- .98871 
15011 
lf,040l 

15069 '.98858 
l.%97  .98854 
15126  .98849 
151551.98845    16878 
16906 


1(899 

15327 
1556.  98814 


15414 

15-142  .9S-00 
15471  .98796 

l."rf)0  9K791 
98787 
15557 


^ p-' 

.     Co.In.  I    M. 


15701 

ir,7.M 

15758 

].->: 

15816 
1684C 

15873 
15902 


16968 

lr('17 
10041 

16074 

16103 
16132 
Kil(X) 
16181 


168ft 
1688 

l»,:j,»; 
n.y,( 
16411 
1644' 
16476 
16505 
.16533 

loia 
16691 

]»;M- 
it;»-7 


16763 
M7M 


17136 
171M 
1719:^ 
17222 
17250  98501 
1727!"  .98496 


80° 


98671 ( 

37 

98€57  36 
98652  35 
98648:  34 
98643!  33 


98619  28 

!«614  27 

98609  26 

98604  25 

18600  24 

98595  28 


OBM 


TABLE   OF   NATURAL    SINES   AND   COSINES. 


TABLE   OF   NATURAL    SIXES    AND   COSINES. 


O»in. 

95iu7; 


159 


M. 

.94552    60 

'.•4542 


..",1040 

95061 

3X694 

.31068 

B6069 

89799 

.31095 

96048 

32749 

.81198 

96088 

89771 

.31151 

95094 

32804 

9:(nr, 

8988S 

!  8190(1 

89869 

.81988 

114997 

89867 

.81961 

919-.- 

89914 

.312-!! 

9-1979 

:-,2942 

.31316 

94970 

32%9 

.81344 

.94961 

.32997 

.31  :,-;•, 

94969 

94943 

[88061 

'.3i':i"7 

9J933 

48079 

.31454 

.'.M924 

88106 

.31482 

.'.'1915 

48184 

.94908 

48161 

181685 

.94897 

.81666 

.94888 

48911 

.81693 

.94878 

.:>••  244 

.31020 

.94860 

48971 

.31C48 

.94861 

.a  2!  s 

.31075 

.31703 

!94841 

;J!J.Vf  .'J; 

.31730 

.94832 

.33381 

.31758 

.94823 

.33408 

.31781 

.94814 

.:::!."( 

.31.-1: 

.94806 

i  :MM':; 

.31841 

.'.  1795 

.88491 

.31808 

.94786 

.88618 

.818% 

!  947  77 

48641 

.31923 

4476E 

18867I 

.31951 

.9475,- 

.3r6(( 

.91719 

;-j;  fj^l 

!  82001 

.94740 

.8E66E 

.890» 

.947^0 

.88685 

.8'  061 

.14721 

*vi7  1  ( 

.89C8! 

.94712 

.aawi 

321  i( 

.94709 

.88764 

!.32144 

.94693 

.33792 

.32171 

.94684 

.33819 

32199 

.94fi74 

.88844 

32227 

94666 

.88874 

89964 

94-  M 

.88901 

32282 

MM, 

88928 

89801 

88961 

89881 

94627 

88981 

89864 

84011 

88883 

.33419 

94599 

'.'.  \(l>it- 

.32447 

94690 

840M 

.32474 

915-n 

.34120 

.89609 

94571 

34147 

.32529 

94661 

31175 

.89667 

94669 

34202 

1  Co>ln.  |    Sine. 

71" 

0«in.  | 

—  TO 

TABLE   OF   NATURAL   SINES   AND  COSINES. 


91355    tt)    ) 


mm  . 

93:253 
30133  .93243 
3616  2 
36 190  i.  93222 
36-217  .93211 
30244  .93^)1 


TABLE   OF    NATURAL   SINES   AND   COSINES. 


161 


.42473 
.42400  .90530  .44072 
..,2525 '.90507  .14-10-1 
.42-552  .9049.-)  .44124 
.42578'. 9048V '.44151 
.43604  .90470  .44177 
.42631  .90458  ,.44203 
.42037  .901  IK  .44220 


28° 

29°       i 

Sine. 

Cwin 

TBST 

c',-r,:\ 

4>i047 

88295 

48481 

vt-tm 

4',!  173 

88281 

1s-:'  t; 

b744s 

J'i'f'.*'.* 

47n24 

S  "*',!(•)" 

8B8M 

16689 

.48557 

>7-io  I 
87420 

47060 

.8BMI 

48688 

87406 

47076 

.88994 

48606 

87891 

.47101 

.88213 

(6684 

87:-:77 

47127 

.88191 

486M 

S7:  f,3 

47168 

.8818C 

.4W184 

.S7340 

4717,s 

>M7- 

.48710 

>I888 

.47204 

.88168 

487% 

xmm 

47999 

>.-H4 

.48761 

.87806 

47966 

>.-!:-;< 

.4b78H 

.87999 

.47281 

>Hr 

.48811 

.>7i7x 

.47306 

.88101 

.48887 

.^72C4 

.47332 

.880b9 

.48862 

.87250 

.47358 

>£07f> 

.46888 

.87235 

.47:,-:; 

8806! 

.48918 

.87991 

[47408 

.earn 

.48888 

.b7i07 

.47434 

>N  ?A 

.48064 

>71!tH 

47460 

.86091 

.48988 

.KI178 

.47488 

.88001 

.41.014 

.87164 

.47511 

.87901 

.49C40 

.87150 

.47681 

.87971 

.4;  i  r,:, 

.H7136 

.4-.B6J 

.8796E 

.49090 

.87121 

.47588 

.Kid 

.49116 

.87107 

.47014 

.87937 

.49141 

.87093 

.47639 

.87925 

.49166 

.87079 

.47660 

.87901 

.49192 

.87064 

.47690 

.87891 

.49217 

.87060 

.47716 

JOOK 

.49242 

.87036 

.47741 

.87868 

.49268 

.87021 

.47767 

.87864 

.49898 

.8T007 

.47793 

.878* 

.49316 

>f.'.,!.3 

47818 

.67891 

.49344 

.£6978 

47844 

jtnstii 

.49869 

.86964 

47860 

.87798 

49804 

.86949 

I.47.-95 

.87784 

.49419 

»  o:;o 

47880 

.8777) 

4944B 

.8K!)21 

.47946 

.87761 

49470 

mn  (i 

.47971 

.87741 

4949C 

86899 

.47997 

.87729 

49521 

86878 

4-122 

.87715 

49646 

SBft  3 

48048 

.K7(H 

.1107! 

M:WO 

48078 
48080 

>7fi^7 
.87672 

4!l5!di 
4%22 

M  S:M 
M>20 

.48124 

.87659 

.49647 

66805 

481CO 

.87645 

49672 

f-1.791 

4,  -17.-. 

.87631 

49697 

K,777 

48901 

.S7f,17 

49799 

Mi762 

48996 

.87C03 

49748 

80748 

4-2.V2 

87689 

407  7:; 

88788 

4  ,-277 

87878 

49798 

».719 

48308 

KT.V,1 

49894 

M1704 

4<:;-S 

87646 

'l!tV  1'* 

B6690 

48354 

87',:  2 

49874 

NiC.75 

48878 

K.Ms 

40699 

vrt-fil 

48406 

87504 

49994 

.-•;t;46 

48480 

87490 

49960 

B6689 

48466 

S747H 

49975 

8rr,i7 

.48461 

87462 

50000 

80608 

Colin.      Sine. 

Cosin.  |    Sine. 

«J 

«)  ) 

50 
58 
57 
56 
55 
54 


I 


10 

9  ? 
8 

1  / 
6 
6  ? 

i:- 

i 

0  ' 


TABLE    OF    NATVRAL    SINES    AND    COSINES. 


Sine. 

50000 

50025 


50126  .86530 
501511.86515 
86501 


30° 


50503 
50588 

50553 
50578 
501103 
501128 
50654 
5')'179 


50S-29 
5IK5! 
50879 


51104 
51129 


5122!) 
51854 


51304 
51329  .85821 


31C 


85926 
85!  ID 


51504 

51690 

51551 

5i  5; '.) 
51(104 
51028 
51653 
51678 
517(1:! 
51728 


518,03 
51X28 
5185-2 
51877 

51902 

51!  I. '7 
51(15-' 

51977 

52002 
53088 


5-2-2  10 
52225 
.52250 
5-2275 

52291) 

52!>t 

5284U 


52  I  IS 


59*  i 


528  I  I 


Cosi 

.a5717 
.85702 
.85687 
.85672 
.85657 
.85ti  J2 


85597 

S558-2 


, 
,85506 


,851;  11 
s.Mlti 
,85431 


85370 


.S5325 

,85310 


.-(KM 


53189 
58314 


53-2H3 
53288 


53730 
53754 

53779 


510-21 
5404'.) 


5  I .'  1  1 
5l'2(i!t 
542: 13 
54317 
,5434-2 
.513IH1 
,54391 
5I115 


.8I5SS 


.811:;:; 
.84417 


.'  I -2!  1-2 
.M-21-, 


33° 

34° 

Sine 

Cosin 

Sine. 

fV^TI. 

.54464 

83807 

55919 

82904 

5  1-188 

83851 

559  13 

82887 

51513 

83835 

55968 

82871 

54537 

83S19 

50992 

.82855 

54561 

83804 

56016 

.82839 

5458i  1 

83788 

5601(1 

54610 

.83772 

.8-J8I  6 

54635 

.8:1756 

'56088 

.82790 

5-165!) 

.837-10 

5611-2 

.82773 

51683 

.83;  21 

.56136 

.82757 

.54708 

..-3708 

.56160 

.82741 

517::-2 

.83(15)2 

.56184 

.82721 

51756 

>:;o:r 

.8-2  ;i>- 

.54781 

'.56232 

.82692 

5-18(15 

'.8SMI 

..-(1256 

.82675 

.54829 

.83629 

.56280 

.82659 

.54854 

.83613 

.56305 

.82643 

.5-1878 

>3:97 

..-(1329 

.821  26 

.54902 

.88681 

.5635:; 

.82010 

.64112! 

.56377 

.51951 

!&354£ 

.56101 

>25  7  7 

.51975 

.56125 

.82661 

.54999 

.83517 

.561-19 

.8251  1 

.55(124 

.83501 

.5617:, 

.8V52X 

.55048 

.56497 

.82511 

.55072 

!8.:i-l(i! 

.56521 

.82495 

.55(197 

.83-15: 

.565  15 

.82-178 

.55121 

.83437 

.5656!) 

.82462 

.•5ii.- 

.8:121 

.5,  15!  13 

.82416 

.83-10: 

.56617 

.82-129 

!55UM 

.83369 

.56641 

.82113 

.55218 

.83373 

.566(15 

.82396 

.66842 

.88361 

.56689 

.82380 

.66801 

.8334C 

.56713 

.82363 

.552!)! 

.6«786 

.82347 

.553ir 

!  83308 

..-,671.0 

.8233d 

.  55339 

.83292 

.56781 

.82314 

.55363 

.8:127  ( 

.82297 

.65888 

.8326C 

:-,6S,2 

.82281 

.55112 

.83241 

.56856 

.8226  1 

8°  "Ms 

.'rci'iu 

.  S.'i^S 
.83212 

.5(i880 
.50901 

1  8223*1 

.55-184 
.6560) 

.83195 
.8317!) 

!  569-28 
.56952 

.82214 

.5553: 

.83163 

.56976 

'.82181 

.55557 

.83147 

.57000 

.82165 

.55581 

.83131 

.57024 

.82148 

.55605 

.83115 

.57047 

.82132 

556:!  i 

>3i  63 

.57071 

.57(195 

.82115 
.82098 

55678 

.57119 

.82182 

55702 

ieaoeo 

.57143 

.82065 

55726 

.83034 

.57167 

.8201-- 

55750 

.83017 

.57191 

.82032 

.55775 

.83001 

.57215 

.82'115 

5579!) 

.82985 

,57238 

.819!)!) 

.55823 

.82969 

TM  "Tl'' 

.81982 

5-8  17 

82963 

.57286 

.819(15 

.55871 

>->!<:i6 

.5731(1 

.81919 

.55*95 

.82920 

.57334 

.81932 

.55919 

.82901 

.57358 

.81915 

Cosin. 

Sine. 

Cosin.   Mne. 

66° 

55° 

19  ( 
18  > 
17  ) 
16  b 
15  ) 
14  \ 
13  ( 
12  ( 
11  } 

10  5 
» 


TABLE    OF    NATURAL    SINKS    AND    COSINES. 


773^9 

.77310  38 
87 

.77273:  36 

.77255!  35 

.77236  34 

.77218  33 

.77199  32 

.771811  31 

.77102:  30 


164 


TABLE    OF    NATURAL    SINKS    AND   COSINSS. 


BY  5.  U.  (SOOMBS. 


PRICE, 


.25. 


CONTENTS  OF  THE  NORMAL  READER. 


INTRODUCTION. 

PART  I. 
How  to  teach  a  child  to  read. 

PART  II. 
Dictionary  Work. 

1.  Pronunciation. 

1.  Key  to  Pronunciation. 

3.  Elementary  Sounds. 

4.  Principles  of  Pronunciation. 

5.  Articulation. 

6.  Words  Often  Mispronounced. 


1.  How  to  Teach  Reading. 

2.  Examples  for  Practice. 


PART  IV. 

Elocution. 
Art  of  Delivery. 
Outline  of  Elocution. 
Plan  of  Studies. 
Elements. 
Respiration. 
Breathing. 
Formulas. 
Articulation. 
Orthoepy. 
Vocal  Culture. 
Exercises  for  Drill. 
Quality 
Vocal 
Volume. 
Rate. 
Gesture. 
Su 

1. 

•2. 


To  Ministers. 
To  Lawyers. 


PART  V. 

One  Hundred  and  Twelve  of  the  best 
Selections  of  Prose  and  Poetry 
from  the  Best  English  and  Amer- 
can  Authors. 


Liberal  terms  for  first  introduction  by  the 
quantity.  Sample  copy  sent  for  $1.00  to  those 
who  desire  to  examine  with  a  view  to  an  adop- 
tion. No  free  copies.  Please  don't  ask  for 
them.  Address, 

J.  E.  SHERR1LL,  Pub., 

DANVILLE,  IND. 


-  ):  DALE'S  :(  - 

OUTLINE  OF  ELOCUTION. 

BY   G.  WALTER 


The  purpose  of  this  book  is  to  afford  a  complete  and  philosoph- 
ical treatment  of  all  the  principles  underlying  the  art  of  human 
expression.  It  is  designed  to.  analyze  the  art  into  its  elements,  and 
discuss  these  elements  in  such  a  manner  as  that  students  who  do 
not  have  the  advantage  of  aid  from  the  living  teacher  may  suc- 
cessfully acquaint  themselves  in  a  practical  manner  with  this  fine 
art.  Its  scope  is  bounded  only  by  the  possibilities  of  the  subject, 
which  makes  it  a  work  of  extraordinary  value,  in  that  it  contains 
the  entire  subject,  logically  treated.  The  definitions,  discussions 
and  exercises  are  concise,  explicit  and  pertinent.  They  are 
adapted  to  students  in  any  grade,  from  the  primary  to  the  high 
school,  seminary  or  college.  The  twelve  appended  Essays  cover 
ground  discussed  in  no  other  similar  work,  and  involve,  among 
others,  "  Care  of  the  Voice,"  "  Primary  Teaching,"  and  a  multi- 
tude of  ;<  Hints  and  Suggestions."  It  is  an  elocutionary  library 
in  itself,  as  it  represents  the  science  and  art  of  expression  by  voice 
and  action.  As  a  book  of  reference  it  is  standard  literature,  and 
a  classic  of  its  kind.  A  student  of  ordinary  intelligence  can  take 
this  book  and  study  elocution  without  the  aid  of  an  instructor, 
which  he  can  not  do  by  the  aid  of  any  other  book.  This  is  be- 
cause the  subject  is  placed  before  the  student  with  such  clearness, 
in  such  minuteness  of  detail,  that  there  is  no  room  for  misunder- 
standing. It  is  the  most  popular  book  published  on  the  subject 
of  Elocution. 

IT  CONTAINS  ESSAYS  ON  : 

1.  Emphasis.  2.  Projection  of  Sound.  3.  Timbre.  4. 
Care  of  the  Toice.  o.  A  Course  of  Reading.  6.  Dramatic 
Reading  and  Recitations.  7.  Impersonation  of  Old  Age. 
8.  Primary  Teaching.  9.  Hints  and  Suggestions,  etc.,  etc. 

The  last  200  pages  are  filled  with  the  best  collection  of  selections 
for  reading  ever  embodied  in  a  book  of  this  kind. 

The  matter  is  new,  the  selections  abundant  and  fresh,  and  the 
tone  of  the  book  immeasurably  above  that  of  the  ordinary  reading 
book.  It  is  the  work  of  one  of  the  most  successful  teachers  of 
reading,  who  is  himself  a  living  example  of  the  high  class  of  in- 
struction he  gives  in  his  book.  Liberal  terms  for  introduction. 
Agents  wanted  all  over  the  world.  The  most  favorable  induce- 
ments offered.  Write  for  terms. 

Address,        J.  E.  SHERRILL,  Pub., 

IPUD. 


THE 


Iijdiarja 


This  is  an  entirely  new  and  original  departure  in 
books  of  this  class,  and  can  not  fail  to  be  a  happy  hit 
and  meet  with  warm  approval  and  a  hearty  welcome 
from  the  public  in  general.  As  its  name  indicates,  it 
is  made  up  of  selections  from  the  very  best  productions 
of  the  very  best  authors  in  Indiana.  We  mean  no 
offense  to  authors  in  limiting  the  work  to  this  State  ; 
neither  do  we  think  it  necessary  to  offer  any  apologies, 
for  her  bright  galaxy  of  distinguished  writers  has  al- 
ready placed  the  "  Hoosier  State  "  in  the  front  rank 
of  the  literary  world.  The  book  contains  writings  of 
much  merit  from  almost  every  class  of  our  educated 
people,  comprising  new,  original,  attractive  and  patri- 
otic recitations'  and  declamations  for  every  grade  of 
pupils.  This  book  will  displace  the  old  and  hack- 
neyed pieces  in  our  school  rooms  by  supplying  orig- 
inal and  the  brightest  and  best  thoughts  and  speeches 
of  our  own  people.  Send  for  a  copy  at  once. 

Address, 

J.E.S^e 

Qa^ville,   Jrjdiarja. 


THF     XORJI.VI. 


DIALOGUE   BOOK. 

PRICE,  FIFTY  CENTS. 

Arranged  with  a  Tien  to  giving  something  highly  entertaining,  and  at  the 
same  time  something  suit  able  and  practicable  for  the  school  exhibition. 


CONTENTS  OF  THE  NORMAL  DIALOGUE  BOOK. 
Miscellaneous. 

Another  Arrangement. 

Aunt  Betsy's  Beau. 

Which  Will  You  Choose? 

Maud's  Command  ;  or,  Yielding  to  Temptation. 

The  Smoke  Fiend. 

Charade.— Phan-Tom. 

Fourth  of  July  Oration. 

A  Woman's  Business  Meeting. 

Boarding  School  Accomplishments. 

Leap  Year  in  the  Village  with  One  Gentleman. 

Too  Greedy  by  Half. 

The  Way  to  Wyndham. 

Little  Folks'  Department. 

Vacation  Fun.  The  Old  Maid. 

The  Secret.  Little  Mischief. 

An  Interrupted  Recitation.         A  Stitch  in  Time  Saves  Nine. 

How  He  Had  Him.  The  Stolen  Pets. 

Tableaux. 

Tableau  1.— Little  Jack  Homer. 

"       2.— The  Old  Woman  who  Lived  in  a  Shoe. 

«       3)  4f  5.— When  I  was  a  Bachelor. 

"       6.— Cinderella. 

«      ?.— Mischief  in  School. 

"       8. — The  Four  Seasons. 

"       9.— The  Witches  in  Macbeth. 

Shadow  Acts  and  Pantomimes. 

Box  and  Cox. 

Courting  Under  Difficulties. 

Hospital  Practice. 

Chapter  of  Instructions. 

Great  Inducements  to  Agents.     Liberal  Terms  to  the  Trade. 

Address,  J.  G,  SffEJWfcfc,  Pub,, 

DANVILJUE,  IND. 


UNIVERSITY  OF  CALIFORNIA  AT  LOS  ANGELES 
THE  UNIVERSITY  LIBRARY 

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